#6557461

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1615 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)(18,19)(21,22)(24,27)(25,26)(28,32)(30,31);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,19,23,26,22,25)(20,28,30,32,29,24,31,27);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,19,22,18,23,25,21,26)(20,27,31,24,29,32,30,28);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(17,20,23,29)(18,24,26,28)(19,27,25,32)(21,30,22,31);(11,15)(14,16)(20,29)(24,28)(27,32)(30,31)

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=fi,d^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6464657);