#7505183

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1537 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=fi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,h^c=hi,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, 7412379);