#6506440

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500434
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35149	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2319 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi

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, 6413636);