#72258295

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#115133
commutator	C₂⁴×C₄	C₂²×C₄
Frattini, cosocle	C₂³×#41	C₂³
prime radical	#6437623	C₂
LCS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂³ ▷ C₂ ▷ C₁

4342 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=f,c^2=gi,d^2=j,b^a=bd,c^a=cde,c^b=cefg,d^a=dj,d^b=dh,d^c=dhi,e^b=ei,e^c=ei,f^a=fhj,f^c=fhi,f^e=fj,g^a=ghj,g^b=gh,g^c=gj,g^e=gj,h^c=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258313, #72258377, #72258453
maximal normal subgroup of	#72258313, #72258377, #72258453

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