#7608610

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1635 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,32)(21,36)(22,28)(23,27)(24,31)(26,35)(29,39)(30,40)(33,43)(34,44)(37,41)(38,45)(42,47)(46,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,33)(21,30)(22,31)(23,37)(25,28)(26,43)(27,40)(29,34)(32,48)(35,46)(36,41)(38,47)(39,45)(42,44);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,19)(18,25)(20,29)(21,37)(22,31)(23,30)(24,28)(26,42)(27,40)(32,39)(33,34)(35,47)(36,41)(38,46)(43,44)(45,48);(10,14)(13,16)(17,20,23,34,22,35,21,38)(18,26,27,45,28,32,36,44)(19,29,30,33,31,47,37,46)(24,39,41,43,25,42,40,48);(2,5)(6,8)(17,21,22,23)(18,27,28,36)(19,30,31,37)(20,34,35,38)(24,40,25,41)(26,44,32,45)(29,46,47,33)(39,43,42,48)

Polycyclic presentation:

2^9|c^2=fi,d^2=h,e^2=i,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,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7515806);