#6700205

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#35737	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1720 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)(11,15)(12,14)(13,16)(17,18)(19,29)(20,25)(21,27)(22,28)(23,26)(24,32)(30,38)(31,45)(33,41)(34,48)(35,42)(36,44)(37,40)(39,46)(43,47);(1,2,4,5)(3,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,30)(21,34)(22,32)(23,37)(25,40)(26,38)(27,43)(28,29)(31,35)(33,46)(36,47)(39,42)(41,45)(44,48);(10,14)(13,15)(17,20)(18,25)(21,35)(22,23)(26,28)(27,42)(30,37)(31,46)(33,36)(38,40)(39,45)(41,44);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20,22,23)(18,26,28,25)(19,30,32,37)(21,33,36,35)(24,38,29,40)(27,42,44,41)(31,34,46,47)(39,48,45,43);(2,5)(6,8)(11,16)(13,15)(17,21,22,36)(18,27,28,44)(19,31,32,46)(20,33,23,35)(24,39,29,45)(25,41,26,42)(30,34,37,47)(38,48,40,43)

Polycyclic presentation:

2^9|b^2=fi,d^2=fg,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,e^c=ei,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6607401);