#6595492

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1368 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:

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

Polycyclic presentation:

2^9|a^2=fhi,c^2=fgi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dhi,e^c=ei,e^d=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6502688);