#6810890

order	512
nilpotency class	3
isoclinism	isoclinic class of #6807031
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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1679 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=f,e^2=fh,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,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, 6718086);