#6668617

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667964
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

2564 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)(11,20)(12,17)(13,28)(14,18)(15,19)(16,22)(21,35)(23,31)(24,34)(25,27)(26,32)(29,33)(30,37)(36,40)(38,39);(1,3,4,8)(2,5,6,7)(9,10)(11,20)(12,25)(13,18)(14,28)(15,19)(16,23)(17,27)(21,30)(22,31)(24,34)(26,33)(29,32)(35,37)(36,39)(38,40);(9,11,15,24)(10,16,19,31)(12,21,26,36)(13,23,14,22)(17,30,32,39)(18,34,28,20)(25,35,33,40)(27,38,29,37);(2,6)(5,7)(12,26)(17,32)(21,36)(25,33)(27,29)(30,39)(35,40)(37,38);(3,8)(5,7)(9,12,15,26)(10,17,19,32)(11,21,24,36)(13,27,14,29)(16,30,31,39)(18,33,28,25)(20,35,34,40)(22,37,23,38)

Polycyclic presentation:

2^9|b^2=f,c^2=i,e^2=i,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,h^c=hi,e^d=ei,h^e=hi

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, 6575813);