#6810411

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806827
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

1899 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)(12,15)(13,18)(14,29)(16,20)(19,32)(21,25)(22,33)(23,37)(24,26)(27,28)(30,40)(31,35)(34,39)(36,38);(1,3,4,8)(2,5,6,7)(9,11,16,25)(10,17,20,21)(12,26,18,33)(13,22,15,24)(14,30,32,38)(19,36,29,40)(23,28,39,31)(27,34,35,37);(3,8)(5,7)(9,12,15,20,16,18,13,10)(11,21,22,26,25,17,24,33)(14,27,31,29,32,35,28,19)(23,37,30,40,39,34,38,36);(2,6)(5,7)(9,13,16,15)(10,18,20,12)(11,22,25,24)(14,28,32,31)(17,33,21,26)(19,35,29,27)(23,30,39,38)(34,36,37,40);(9,14,15,31,16,32,13,28)(10,19,12,27,20,29,18,35)(11,23,22,30,25,39,24,38)(17,34,33,36,21,37,26,40)

Polycyclic presentation:

2^9|b^2=fi,c^2=fh,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, 6717607);