#356819

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1258 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=egi,d^2=eg,f^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,d^b=de,f^c=fi

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, 264015);