#7609547

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	#45093	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1844 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)(17,18,22,20)(19,27,28,24)(21,30,29,25)(23,26,32,31);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,28)(18,24,20,27)(21,32,29,23)(25,31,30,26);(10,14)(11,15)(17,20,22,18)(19,27,28,24)(21,30,29,25)(23,31,32,26);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,21,23,19,22,29,32,28)(18,25,26,24,20,30,31,27);(2,5)(6,8)(18,20)(24,27)(25,30)(26,31)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fhi,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

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, 7516743);