#6671987

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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2066 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)(11,17)(12,25)(13,19)(14,20)(15,32)(16,21)(18,27)(22,34)(23,35)(24,36)(26,33)(28,39)(29,38)(30,40)(31,37);(1,2,4,5)(3,6,7,8)(9,11)(10,17)(12,26)(13,22)(14,28)(15,23)(16,24)(18,37)(19,34)(20,39)(21,36)(25,33)(27,31)(29,30)(32,35)(38,40);(9,12,16,29)(10,18,21,39)(11,14,24,31)(13,27,15,28)(17,33,36,40)(19,38,32,25)(20,34,37,35)(22,26,23,30);(1,3,4,7)(2,6,5,8)(9,13,16,15)(10,19,21,32)(11,22,24,23)(12,27,29,28)(14,26,31,30)(17,34,36,35)(18,38,39,25)(20,33,37,40);(2,5)(6,8)(9,14,16,31)(10,20,21,37)(11,12,24,29)(13,30,15,26)(17,25,36,38)(18,34,39,35)(19,40,32,33)(22,28,23,27)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=fi,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, 6579183);