#7510269

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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1267 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)(3,4)(5,7)(6,8)(9,10,13,15)(11,14,12,16)(17,18)(19,28)(20,33)(21,37)(22,27)(23,26)(24,31)(25,36)(29,39)(30,40)(32,44)(34,41)(35,42)(38,43)(45,46)(47,48);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,34)(21,32)(22,31)(23,30)(25,41)(26,44)(27,28)(29,35)(33,48)(36,47)(37,40)(38,46)(39,43)(42,45);(9,12,13,11)(10,14,15,16)(17,20,22,36)(18,25,27,33)(19,29,31,46)(21,35,23,38)(24,39,28,45)(26,42,37,43)(30,34,32,47)(40,48,44,41);(2,7)(3,8)(10,15)(14,16)(17,21,22,23)(18,26,27,37)(19,30,31,32)(20,35,36,38)(24,40,28,44)(25,42,33,43)(29,34,46,47)(39,48,45,41);(1,4,5,6)(2,3,7,8)(18,27)(19,31)(25,33)(26,37)(29,46)(30,32)(34,47)(42,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi

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, 7417465);