#357089

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

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

1272 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)(9,10)(11,14)(12,13)(15,16)(17,18,21,24,22,26,20,25)(19,23,29,30,27,31,28,32);(5,6,7,8)(10,13)(11,15)(17,19)(18,23)(20,28)(21,29)(22,27)(24,30)(25,32)(26,31);(1,2,4,3)(18,24)(20,21)(23,30)(25,26)(28,29)(31,32);(1,3,4,2)(9,11,12,15)(10,14,13,16)(19,27)(23,31)(28,29)(30,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=efg,f^2=i,g^2=i,b^a=be,c^a=cf,d^b=dei,f^c=fi,g^c=gi

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, 264285);