#6684994

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

1897 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,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, 6592190);