#6806832

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

1940 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)(11,16)(12,13)(14,17)(15,19)(18,28)(20,24)(21,25)(22,32)(23,33)(26,27)(29,34)(30,40)(31,35)(36,38)(37,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,25)(13,21)(14,29)(15,24)(17,34)(18,33)(19,20)(22,27)(23,28)(26,32)(30,36)(31,37)(35,39)(38,40);(9,12,13,19,15,18,28,10)(11,20,23,25,24,16,21,33)(14,30,27,17,31,26,40,35)(22,32,29,34,38,36,37,39);(2,6)(5,7)(9,13,15,28)(10,12,19,18)(11,21,24,23)(14,27,31,40)(16,25,20,33)(17,26,35,30)(22,37,38,29)(32,39,36,34);(3,8)(5,7)(9,14,28,40,15,31,13,27)(10,17,18,30,19,35,12,26)(11,22,21,37,24,38,23,29)(16,32,25,39,20,36,33,34)

Polycyclic presentation:

2^9|c^2=h,d^2=i,e^2=hi,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,e^c=ei,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, 6714028);