#6809307

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806827
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

2209 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,24)(13,17)(14,28)(15,19)(18,30)(20,31)(21,37)(22,25)(23,33)(26,27)(29,40)(32,39)(34,38)(35,36);(2,5)(3,7)(9,11,15,23)(10,16,19,33)(12,25,17,31)(13,20,24,22)(14,29,30,36)(18,35,28,40)(21,38,39,27)(26,37,34,32);(9,12,13,10,15,17,24,19)(11,16,22,25,23,33,20,31)(14,26,27,18,30,34,38,28)(21,32,29,35,39,37,36,40);(2,5)(6,8)(9,13,15,24)(10,17,19,12)(11,20,23,22)(14,27,30,38)(16,31,33,25)(18,34,28,26)(21,36,39,29)(32,40,37,35);(1,3,4,7)(2,6,5,8)(9,14,13,27,15,30,24,38)(10,18,17,34,19,28,12,26)(11,21,22,29,23,39,20,36)(16,32,25,35,33,37,31,40)

Polycyclic presentation:

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

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, 6716503);