#354977

order	512
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×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×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1186 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)(11,19)(12,16)(13,25)(14,18)(15,22)(17,26)(20,31)(21,32)(23,36)(24,30)(27,34)(28,35)(29,37)(33,39)(38,40);(9,11)(10,15,18,27)(12,21)(13,22)(14,20)(16,28,30,38)(17,19,25,31)(23,35)(24,33)(26,34)(29,32,36,39)(37,40);(1,2,3,4)(11,20)(15,27)(19,31)(21,33)(22,34)(28,38)(32,39)(35,40);(9,12,13,23,14,24,26,37)(10,16,25,36,18,30,17,29)(11,21,22,35,20,33,34,40)(15,28,31,39,27,38,19,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,d^2=eh,e^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,c^b=ci

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210536
maximal normal subgroup of	#72210536
maximal quotient of	#72210396

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262173);