#6462660

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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×#469	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1424 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=f,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6369856);