#6683768

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682648
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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2212 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=fi,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,d^c=di,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, 6590964);