#492857

order	512
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂³	C₂²×D₈
prime radical	C₂⁵	D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1698 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319878, #72319893, #72319934, …
maximal normal subgroup of	#72319878, #72319893, #72319934, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400053);