#7618040

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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1242 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fi,d^2=h,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,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, 7525236);