#507609

order	512
nilpotency class	4
isoclinism	isoclinic class of #499891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

488 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)(18,22)(19,25)(21,29)(23,31)(24,28)(26,30)(27,32);(1,3,5,4)(2,6,8,7)(9,10,12,14)(11,15,16,13)(17,18,20,22)(19,26,27,23)(21,24,28,29)(25,31,32,30);(9,11,12,16)(10,13,14,15)(17,19,20,27)(18,23,22,26)(21,30,28,31)(24,32,29,25);(1,4,5,3)(2,7,8,6)(17,18,20,22)(19,23,27,26)(21,24,28,29)(25,30,32,31)

Polycyclic presentation:

2^9|a^2=ei,b^2=egi,c^2=g,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,f^c=fh,h^c=hi

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, 414805);