#6450673

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1576 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:

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

Polycyclic presentation:

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

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, 6357869);