#472068

order	512
nilpotency class	3
isoclinism	isoclinic class of #454382
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1204 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=eghi,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379264);