#495025

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

682 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)(11,19)(12,20)(13,23)(15,25)(17,26)(18,30)(21,35)(22,33)(24,34)(27,36)(28,40)(29,39)(31,32)(37,38);(2,4)(5,7,8,6)(9,10)(11,15)(12,17)(13,18)(14,16)(19,25)(20,26)(21,28)(22,27)(23,30)(24,29)(31,37)(32,38)(33,36)(34,39)(35,40);(9,11,20,32,13,21,24,33,14,22,34,35,23,31,12,19)(10,15,26,38,18,28,29,36,16,27,39,40,30,37,17,25);(10,16)(15,27)(17,29)(18,30)(25,36)(26,39)(28,37)(38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fh,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,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, 402221);