#6816966

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814827
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

2094 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=f,e^2=fhi,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,e^c=ei

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, 6724162);