#495497

order	512
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	C₂×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	#3444	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

852 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)(3,6)(4,5)(7,8)(9,10,13,18,14,17,27,30,15,19,28,33,12,26,21,36)(11,20,24,35,25,31,38,39,22,34,37,40,23,16,29,32);(9,11,21,29,12,23,28,37,15,22,27,38,14,25,13,24)(10,16,30,35,26,34,18,32,19,31,36,40,17,20,33,39);(2,5)(6,8)(11,22)(16,31)(20,34)(23,25)(24,37)(29,38)(32,39)(35,40);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402693);