#369490

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2361
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1042 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276686);