#7596279

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	#45135	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=fi,c^2=fh,d^2=fgi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=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, 7503475);