#7535072

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45267	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1868 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)(11,15)(12,14)(13,16)(17,18,22,27)(19,28,32,24)(20,25,34,43)(21,35,23,26)(29,45,47,38)(30,46,48,39)(31,40,37,41)(33,44,36,42);(1,3,5,4)(2,6,8,7)(11,16)(13,15)(17,19,22,32)(18,24,27,28)(20,29,34,47)(21,31,23,37)(25,38,43,45)(26,40,35,41)(30,36,48,33)(39,42,46,44);(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,29)(21,36)(22,34)(23,33)(24,38)(26,44)(27,43)(28,45)(30,37)(31,48)(32,47)(35,42)(39,40)(41,46);(1,3,5,4)(2,7,8,6)(10,14)(13,15)(17,20)(18,25)(19,30)(21,33)(22,34)(23,36)(24,39)(26,42)(27,43)(28,46)(29,37)(31,47)(32,48)(35,44)(38,40)(41,45);(1,4,5,3)(2,7,8,6)(17,21,22,23)(18,26,27,35)(19,31,32,37)(20,33,34,36)(24,40,28,41)(25,42,43,44)(29,30,47,48)(38,46,45,39)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7442268);