#7603551

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1362 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(9,10,13,12)(11,14,16,15)(17,18)(19,27)(20,31)(21,23)(22,35)(24,28)(25,32)(26,33)(29,44)(30,38)(34,40)(36,43)(37,46)(39,45)(41,42)(47,48);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(18,23)(19,28)(20,32)(22,33)(24,38)(25,40)(27,36)(29,45)(30,43)(31,37)(34,46)(41,47);(1,3,5,4)(2,7,8,6)(9,12,13,10)(11,15,16,14)(17,19,22,28,21,30,33,43)(18,24,26,27,23,36,35,38)(20,29,32,42,34,45,46,48)(25,39,31,41,37,44,40,47);(1,4,5,3)(2,6,8,7)(11,16)(14,15)(17,20,33,46,21,34,22,32)(18,25,35,40,23,37,26,31)(19,29,43,48,30,45,28,42)(24,39,38,47,36,44,27,41);(2,8)(6,7)(18,23)(19,30)(25,37)(26,35)(28,43)(29,45)(31,40)(42,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=fi,c^2=fhi,d^2=fg,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,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, 7510747);