#6806819

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806815
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

1924 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=h,d^2=i,e^2=h,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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6714015);