#6448243

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1448 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=fgi,d^2=f,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, 6355439);