#72238495

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23984
commutator, Frattini, cosocle	C₂²×C₂²⋊C₄	C₂⁴
prime radical	#346449	C₂
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#121 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

2953 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:

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

Polycyclic presentation:

2^10|a^2=g,c^2=j,d^2=j,f^2=ij,c^b=cghij,d^a=df,d^c=de,e^a=eh,e^b=ei,f^c=fhj,f^d=fij,f^e=fj,g^d=gij,h^d=hj

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238499, #72262594, #72262632
maximal normal subgroup of	#72238499, #72262594, #72262632

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