#507501

order	512
nilpotency class	4
isoclinism	isoclinic class of #501046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

510 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=egi,c^2=fh,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414697);