#497692

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₂²⋊C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

704 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)(3,5)(4,6)(7,8)(9,10,14,19,13,20,23,39,15,21,28,40,31,18,36,34)(11,22,25,35,26,33,29,38,27,37,12,17,24,16,30,32);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,28)(13,26)(14,25)(15,27)(17,34)(18,37)(19,32)(20,22)(21,33)(23,29)(24,31)(30,36)(35,39)(38,40);(2,6)(5,8);(9,12)(10,17)(11,23)(13,30)(14,24)(15,25)(16,19)(18,38)(20,32)(21,35)(22,39)(26,28)(27,36)(29,31)(33,40)(34,37)

Polycyclic presentation:

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

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, 404888);