#6693589

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1326 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)(19,28)(20,33)(21,27)(22,25)(23,26)(24,31)(29,38)(30,45)(32,41)(34,44)(35,48)(36,40)(37,42)(39,47)(43,46);(1,3,5,4)(2,6,7,8)(9,11,13,10)(12,15,16,14)(17,19,22,31)(18,24,25,28)(20,32,23,29)(21,35,36,46)(26,41,33,38)(27,43,40,48)(30,34,47,37)(39,44,45,42);(1,4,5,3)(2,6,7,8)(10,11)(14,15)(18,25)(19,29)(20,23)(24,38)(27,40)(28,41)(30,46)(31,32)(34,37)(35,47)(39,43)(45,48);(9,12,13,16)(10,14,11,15)(17,20,22,23)(18,26,25,33)(19,29,31,32)(21,34,36,37)(24,38,28,41)(27,42,40,44)(30,46,47,35)(39,43,45,48);(17,21,22,36)(18,27,25,40)(19,30,31,47)(20,34,23,37)(24,39,28,45)(26,42,33,44)(29,46,32,35)(38,43,41,48)

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=fg,d^2=f,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, 6600785);