#6693553

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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1346 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=fg,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,d^c=di,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, 6600749);