#6671992

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667964
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

2074 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=f,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,g^c=gi,h^c=hi,e^d=ei,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, 6579188);