#7611871

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563337
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

1912 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=f,d^2=h,e^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7519067);