#356884

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,23,19,24,17,20)(15,21,27,35,28,36,26,32)(16,22,29,38,30,37,31,33)(25,34,39,44,40,43,41,42);(1,2)(3,4)(14,20,24,23)(16,29,30,31)(21,32,36,35)(22,37)(25,39,40,41)(33,38)(34,43)(42,44);(6,9)(10,12)(13,15,18,27,19,28,17,26)(14,21,23,35,24,36,20,32)(16,25,29,39,30,40,31,41)(22,34,38,44,37,43,33,42);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,29,19,30,17,31)(14,22,23,38,24,37,20,33)(15,25,27,39,28,40,26,41)(21,34,35,44,36,43,32,42)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=fg,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=de,e^d=ei,h^d=hi

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, 264080);