#498328

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
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	C₂²×D₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

668 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405524);