#72262706

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#21746
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	#4380225	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#534 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

4088 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:

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

Polycyclic presentation:

2^10|a^2=hi,b^2=ef,e^2=j,b^a=be,c^a=cg,c^b=cg,d^a=dgj,d^b=dij,e^b=eh,e^c=ei,e^d=ei,f^a=fhj,f^b=fh,f^d=fi,f^e=fj,g^b=gi,h^b=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262707, #72262713, #72262719
maximal normal subgroup of	#72262707, #72262713, #72262719

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