#7620071

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1223 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)(11,16)(12,15)(13,14)(17,18)(19,27)(20,25)(21,23)(22,26)(24,28)(29,40)(30,41)(31,39)(32,34)(33,35)(36,42)(37,43)(38,46)(44,47)(45,48);(1,3,5,4)(2,6,8,7)(9,10)(11,16)(12,15)(13,14)(18,23)(19,28)(20,31)(22,33)(25,38)(29,44)(32,46)(34,39)(36,47)(37,41)(40,48)(42,45);(1,3,5,4)(2,7,8,6)(10,14)(15,16)(17,19)(18,24)(20,29)(21,28)(22,30)(23,27)(25,36)(26,37)(31,45)(32,42)(33,43)(34,40)(35,41)(38,48)(39,47)(44,46);(9,11,13,12)(10,15,14,16)(17,20,22,31,21,32,33,46)(18,25,26,39,23,34,35,38)(19,29,30,45,28,42,43,44)(24,36,37,47,27,40,41,48);(1,4,5,3)(2,7,8,6)(9,12,13,11)(10,15,14,16)(18,23)(19,28)(25,34)(26,35)(29,42)(30,43)(38,39)(44,45)

Polycyclic presentation:

2^9|b^2=fi,c^2=fi,d^2=h,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,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, 7527267);