#6450710

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1540 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)(19,29)(20,26)(21,27)(22,24)(23,28)(25,33)(30,38)(31,43)(32,40)(34,42)(35,37)(36,41)(39,46)(44,47)(45,48);(1,3,4,8)(2,5,6,7)(10,13)(14,16)(18,24)(19,30)(20,34)(21,23)(25,38)(26,41)(29,40)(32,33)(35,36)(37,42)(39,43)(44,45);(10,13)(14,16)(17,19,22,33)(18,25,24,29)(20,31,35,46)(21,30,23,32)(26,39,37,43)(27,40,28,38)(34,44,36,45)(41,47,42,48);(9,11,12,15)(10,14,13,16)(17,20,22,35)(18,26,24,37)(19,31,33,46)(21,34,23,36)(25,39,29,43)(27,42,28,41)(30,44,32,45)(38,47,40,48);(17,21,22,23)(18,27,24,28)(19,32,33,30)(20,34,35,36)(25,38,29,40)(26,42,37,41)(31,45,46,44)(39,47,43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6357906);