#6450236

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1707 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)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,15,16)(17,18)(19,29)(20,35)(21,26)(22,40)(23,28)(24,27)(25,33)(30,41)(31,47)(32,42)(34,46)(36,43)(37,44)(38,48)(39,45);(2,5)(6,8)(17,19,23,33)(18,25,28,29)(20,36,21,38)(22,34,24,32)(26,43,35,48)(27,46,40,42)(30,39,31,37)(41,44,47,45);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,23,21)(18,26,28,35)(19,30,33,31)(22,37,24,39)(25,41,29,47)(27,44,40,45)(32,36,34,38)(42,48,46,43);(9,11,12,15)(10,14,13,16)(17,21,23,20)(18,26,28,35)(19,31,33,30)(22,39,24,37)(25,41,29,47)(27,44,40,45)(32,38,34,36)(42,48,46,43);(17,22,23,24)(18,27,28,40)(19,32,33,34)(20,37,21,39)(25,42,29,46)(26,44,35,45)(30,36,31,38)(41,48,47,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=f,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,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, 6357432);