#6816450

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1720 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)(3,7)(4,6)(5,8)(9,10)(11,20)(12,16)(13,18)(14,19)(15,24)(17,27)(21,22)(23,33)(25,28)(26,31)(29,36)(30,38)(32,39)(34,35)(37,40);(1,3,4,8)(2,5,6,7)(9,10)(11,21)(12,25)(13,17)(14,19)(15,24)(16,28)(18,27)(20,22)(23,37)(26,32)(29,36)(30,35)(31,39)(33,40)(34,38);(2,6)(3,8)(9,11,14,22)(10,15)(12,23,26,35)(13,24,27,36)(16,30)(17,20)(18,21)(19,29)(25,34)(28,38,39,40)(31,37)(32,33);(2,6)(5,7)(11,22)(12,26)(15,29)(16,31)(20,21)(24,36)(25,32)(28,39);(9,12,27,39,14,26,13,28)(10,16,17,32,19,31,18,25)(11,23,36,40,22,35,24,38)(15,30,20,33,29,37,21,34)

Polycyclic presentation:

2^9|b^2=f,c^2=gh,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^d=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, 6723646);