#7601658

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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1308 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gh,d^2=g,e^2=fgi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di

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, 7508854);