#7603052

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1363 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,23)(20,33)(21,26)(22,28)(24,31)(25,36)(27,35)(29,42)(30,39)(32,40)(34,38)(37,43)(41,46)(44,47)(45,48);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,34)(21,30)(22,32)(24,27)(25,42)(26,39)(28,40)(29,36)(31,35)(33,38)(37,45)(41,44)(43,48)(46,47);(10,14)(11,16)(17,19,21,31,22,32,35,30)(18,24,27,23,28,39,26,40)(20,29,36,34,37,46,47,45)(25,41,33,38,44,42,43,48);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20,35,47,22,37,21,36)(18,25,26,43,28,44,27,33)(19,29,30,45,32,46,31,34)(23,38,24,41,40,48,39,42);(2,5)(6,8)(10,14)(13,15)(17,21,22,35)(18,26,28,27)(19,30,32,31)(20,36,37,47)(23,39,40,24)(25,43,44,33)(29,45,46,34)(38,42,48,41)

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=fgi,e^2=i,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, 7510248);