#7609806

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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	#45105	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1856 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(19,26)(23,31)(27,32)(28,30);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,27)(20,24)(21,29)(22,25)(23,28)(26,32)(30,31);(10,14)(11,15)(17,18)(19,23)(20,29)(21,24)(22,25)(26,31)(27,28)(30,32);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,19,21,28,22,26,20,30)(18,23,24,27,25,31,29,32);(2,7)(6,8)(17,20,22,21)(18,24,25,29)(19,28,26,30)(23,32,31,27)

Polycyclic presentation:

2^9|b^2=f,d^2=fh,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=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, 7517002);