#6750077

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#39471	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1740 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257462, #72257577, #72264193
maximal normal subgroup of	#72257462, #72257577, #72264193
maximal quotient of	#72307868

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6657273);