#6449335

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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

1616 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,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, 6356531);