#384181

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

1334 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)(9,10)(11,15)(12,14)(13,16)(17,18,22,23,21,25,19,26)(20,24,29,30,28,31,27,32);(1,3,4,2)(5,6)(7,8)(17,19,21,22)(18,23,25,26)(20,27,28,29)(24,30,31,32);(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32);(2,3)(10,14)(13,15)(20,28)(24,31)(27,29)(30,32)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=dei,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 291377);