#6718518

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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1648 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)(9,10,12,13)(11,14,15,16)(17,18,22,27)(19,29,33,24)(20,25,36,43)(21,26,23,28)(30,45,47,38)(31,40,32,39)(34,48,46,41)(35,42,37,44);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19,22,33)(18,24,27,29)(20,34,36,46)(21,32,23,31)(25,41,43,48)(26,40,28,39)(30,37,47,35)(38,44,45,42);(10,13)(14,16)(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,38)(26,42)(27,43)(28,44)(29,45)(31,34)(32,46)(33,47)(39,41)(40,48);(2,6)(5,7)(10,13)(14,16)(17,21,22,23)(18,26,27,28)(19,31,33,32)(20,35,36,37)(24,39,29,40)(25,42,43,44)(30,34,47,46)(38,41,45,48);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(19,32)(21,23)(24,40)(26,28)(29,39)(30,46)(31,33)(34,47)(35,37)(38,48)(41,45)(42,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,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, 6625714);