#7621318

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

1264 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)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,30)(20,34)(21,26)(22,29)(23,28)(24,32)(25,38)(27,40)(31,42)(33,47)(35,43)(36,46)(37,45)(39,48)(41,44);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,32)(18,24,28,30)(20,35,38,31)(21,40,33,22)(25,43,34,42)(26,29,47,27)(36,37,41,39)(44,45,46,48);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,23,38)(18,25,28,34)(19,31,32,35)(21,41,33,36)(22,39,40,37)(24,42,30,43)(26,46,47,44)(27,48,29,45);(10,13)(14,16)(17,21,22,19,23,33,40,32)(18,26,29,30,28,47,27,24)(20,36,39,35,38,41,37,31)(25,44,45,42,34,46,48,43);(1,4,5,3)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,22,23,40)(18,27,28,29)(19,21,32,33)(20,37,38,39)(24,26,30,47)(25,45,34,48)(31,36,35,41)(42,44,43,46)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=ghi,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,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, 7528514);