#6458571

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1831 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)(3,7)(4,8)(5,6)(9,10,13,15)(11,14,12,16)(17,18,22,28,23,29,24,30)(19,31,33,26,34,40,20,25)(21,27,36,43,37,44,38,45)(32,46,47,41,48,42,35,39);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,23,34)(18,25,29,26)(20,24,33,22)(21,35,37,47)(27,42,44,46)(28,40,30,31)(32,36,48,38)(39,43,41,45);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,20,24,34,23,33,22,19)(18,26,30,31,29,25,28,40)(21,35,38,48,37,47,36,32)(27,41,45,46,44,39,43,42);(1,4,5,3)(2,8,6,7)(17,21,22,36,23,37,24,38)(18,27,28,43,29,44,30,45)(19,32,33,47,34,48,20,35)(25,39,31,46,26,41,40,42);(17,22,23,24)(18,28,29,30)(19,20,34,33)(21,36,37,38)(25,40,26,31)(27,43,44,45)(32,35,48,47)(39,42,41,46)

Polycyclic presentation:

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

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, 6365767);