#6678109

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	#35757	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2255 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,25)(19,27,29,23)(20,24,32,39)(22,26,33,40)(28,41,34,36)(30,43,45,37)(31,38,35,42)(44,48,46,47);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(19,28)(20,31)(22,33)(23,36)(24,38)(26,40)(27,41)(29,34)(32,35)(39,42)(44,46)(47,48);(10,14)(13,16)(17,19)(18,23)(20,30)(21,29)(22,34)(24,37)(25,27)(26,41)(28,33)(31,44)(32,45)(35,46)(36,40)(38,47)(39,43)(42,48);(2,6)(5,7)(19,29)(23,27)(28,34)(30,45)(36,41)(37,43)(44,46)(47,48);(3,8)(5,7)(17,20,21,32)(18,24,25,39)(19,30,29,45)(22,35,33,31)(23,37,27,43)(26,42,40,38)(28,44,34,46)(36,47,41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,e^2=i,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, 6585305);