#6684988

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682658
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592184);