#495822

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂×#2206	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

784 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403018);