#72251910

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1898
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂⁴
prime radical	#619125	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#525 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

4137 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^10|c^2=f,d^2=j,e^2=ij,b^a=bg,c^a=ci,c^b=ce,d^b=di,e^a=ei,e^b=eij,e^c=eh,f^b=fhij,f^e=fj,g^c=gi,h^c=hj

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251912, #72251996, #72252028
maximal normal subgroup of	#72251912, #72251996, #72252028

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