#6733305

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#30502	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1549 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=fg,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6640501);