#373995

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

1034 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=h,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281191);