#7609496

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45103	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1879 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7516692);