#6458356

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1628 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18,23,29,24,30,22,28)(19,31,35,25,36,42,34,43)(20,26,38,45,37,44,21,27)(32,47,33,40,39,41,48,46);(2,6)(7,8)(17,19,24,36)(18,25,30,43)(20,33,37,48)(21,39,38,32)(22,35,23,34)(26,41,44,47)(27,46,45,40)(28,42,29,31);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20,23,38,24,37,22,21)(18,26,29,45,30,44,28,27)(19,32,35,33,36,39,34,48)(25,40,42,41,43,46,31,47);(1,4,5,3)(2,8,6,7)(17,21,22,37,24,38,23,20)(18,27,28,44,30,45,29,26)(19,33,34,32,36,48,35,39)(25,41,31,40,43,47,42,46);(20,37)(21,38)(26,44)(27,45)(32,39)(33,48)(40,46)(41,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=fgi,d^2=f,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,e^d=ei

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, 6365552);