#7603631

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

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=cgi,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, 7510827);