#6693611

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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)(20,26)(21,27)(22,25)(23,28)(29,38)(30,39)(31,40)(32,37)(33,42)(34,43)(35,41)(36,44)(45,48)(46,47);(1,3,5,4)(2,6,7,8)(9,11,13,10)(12,15,16,14)(17,19,22,31)(18,24,25,40)(20,32,23,29)(21,34,35,46)(26,37,28,38)(27,43,41,47)(30,36,45,33)(39,44,48,42);(1,4,5,3)(2,6,7,8)(10,11)(14,15)(18,25)(19,29)(20,23)(24,37)(27,41)(30,34)(31,32)(33,36)(38,40)(39,47)(43,48)(45,46);(9,12,13,16)(10,14,11,15)(17,20,22,23)(18,26,25,28)(19,29,31,32)(21,33,35,36)(24,38,40,37)(27,42,41,44)(30,34,45,46)(39,43,48,47);(17,21,22,35)(18,27,25,41)(19,30,31,45)(20,33,23,36)(24,39,40,48)(26,42,28,44)(29,34,32,46)(37,47,38,43)

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=fgi,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,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, 6600807);