#7620476

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1281 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fi,d^2=fgh,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7527672);