#72228008

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁵
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

12755 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

(1,2)(3,7)(4,6)(5,8)(11,12)(13,22)(15,19)(18,23);(1,3,4,8)(2,5,6,7)(10,15)(11,12)(16,23)(17,19)(18,24)(20,21);(2,6)(5,7)(9,10,13,18,14,17,22,23)(11,19,20,24,12,15,21,16);(2,6)(5,7)(9,11,14,12)(10,16,17,24)(13,21,22,20)(15,23,19,18);(3,8)(5,7)(10,17)(11,12)(18,23)(20,21)

Polycyclic presentation:

2^10|b^2=hij,c^2=g,d^2=j,f^2=j,g^2=j,b^a=bh,c^a=chj,c^b=cf,d^a=dhj,d^b=di,d^c=dg,e^b=eh,e^c=ej,e^d=ej,f^a=fj,f^b=fj,f^c=fi,f^d=fij,f^e=fj,g^a=gj,g^b=gij,g^d=gj,g^f=gj,h^c=hj,i^c=ij

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72228023, #72228027
maximal subgroup of	#72228015, #72228023, #72324336
maximal normal subgroup of	#72228015, #72228023, #72324336
maximal quotient of	#72268023, #72324340

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.8*F.9*F.10);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);