#494910

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=e,d^2=i,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^b=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, 402106);