#357016

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1492 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,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, 264212);