#511508

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

487 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=efi,b^2=ei,c^2=e,d^2=efi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,h^b=hi,g^c=gh,h^c=hi

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, 418704);