#72232878

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#392135
commutator	C₂²×D₄	C₂³×C₄
Frattini, cosocle	C₂×H(5,𝔽₂)	C₂⁴
prime radical	#10586817	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂×H(5,𝔽₂) ▷ C₂² ▷ C₂ ▷ C₁

At least 17087 conjugacy classes of subgroups; only the 689 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(3,5)(8,12)(14,19)(15,30)(17,29)(23,27)(24,31)(25,32);(1,2)(3,12)(4,9)(5,8)(6,20)(7,11)(10,21)(13,28)(14,23)(15,31)(16,22)(17,25)(18,26)(19,27)(24,30)(29,32);(1,3,7,5)(2,8,11,12)(4,15)(6,14,21,19)(9,24)(10,23,20,27)(13,29)(16,17)(18,30)(22,32)(25,28)(26,31);(1,4,5,16,7,18,3,13)(2,9,8,22,11,26,12,28)(6,17,14,15,21,29,19,30)(10,25,27,31,20,32,23,24)

Polycyclic presentation:

2^10|c^2=i,d^2=e,e^2=j,f^2=j,c^a=cj,c^b=cj,d^a=dh,d^b=dg,d^c=df,e^a=ei,e^c=eij,f^a=fij,f^b=fi,f^d=fi,f^e=fj,g^c=gij,g^f=gj,h^c=hi,h^d=hi,h^e=hj,h^f=hj,i^d=ij

Conjugacy classes & character table

91 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260582
maximal normal subgroup of	#72260582
maximal quotient of	#72232927, #72233346, #72238428

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, 3, F.9);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);