#7615794

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1358 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)(11,15)(12,14)(13,16)(17,18,23,28)(19,24,31,40)(20,25,36,45)(21,26,22,27)(29,38,33,42)(30,39,37,46)(32,41,47,48)(34,43,35,44);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,32)(21,30)(22,37)(23,31)(25,41)(26,39)(27,46)(28,40)(29,35)(33,34)(36,47)(38,44)(42,43)(45,48);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,19)(18,24)(20,33)(21,37)(22,30)(23,31)(25,42)(26,46)(27,39)(28,40)(29,36)(32,34)(35,47)(38,45)(41,43)(44,48);(10,14)(13,15)(17,20,22,35,23,36,21,34)(18,25,27,44,28,45,26,43)(19,29,30,32,31,33,37,47)(24,38,39,41,40,42,46,48);(2,5)(6,8)(10,14)(13,15)(17,21,23,22)(18,26,28,27)(19,30,31,37)(20,34,36,35)(24,39,40,46)(25,43,45,44)(29,32,33,47)(38,41,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fg,d^2=h,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 7522990);