#369492

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
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

1106 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)(11,15)(12,13)(14,16)(17,18,22,25,21,24,26,19)(20,28,31,27,30,23,32,29);(1,2,4,3)(6,8)(10,13)(14,15)(17,19,26,24,21,25,22,18)(20,29,32,23,30,27,31,28);(1,2,4,3)(5,6,7,8)(17,20)(18,23)(19,27)(21,30)(22,31)(24,28)(25,29)(26,32);(1,3,4,2)(9,11,12,16)(10,14,13,15)(20,30)(23,28)(27,29)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276688);