#494912

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

804 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=e,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,h^a=hi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402108);