#7611868

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563333
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

1964 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)(9,10,16,20)(11,21,19,15)(12,24,29,17)(13,30,31,18)(14,32,34,23)(22,28,25,37)(26,35,40,36)(27,39,33,38);(1,3)(2,5)(4,7)(6,8)(9,11,15,10,16,19,21,20)(12,25,28,24,29,22,37,17)(13,30,14,23,31,18,34,32)(26,35,33,39,40,36,27,38);(2,6)(3,7)(9,12)(10,17)(11,22)(13,26)(14,33)(15,28)(16,29)(18,35)(19,25)(20,24)(21,37)(23,38)(27,34)(30,36)(31,40)(32,39);(2,6)(5,8)(9,13,15,14,16,31,21,34)(10,18,11,23,20,30,19,32)(12,26,28,33,29,40,37,27)(17,35,22,38,24,36,25,39);(9,14,21,13,16,34,15,31)(10,18,11,23,20,30,19,32)(12,27,37,40,29,33,28,26)(17,36,22,39,24,35,25,38)

Polycyclic presentation:

2^9|a^2=i,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,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, 7519064);