#6722410

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	#36015	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1568 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)(11,16)(12,14)(13,15)(17,18,22,26)(19,23,29,38)(20,24,31,40)(21,25,35,44)(27,36,45,47)(28,37,33,42)(30,39,46,48)(32,41,34,43);(1,2)(3,6)(4,5)(7,8)(17,19,22,29)(18,23,26,38)(20,30,31,46)(21,33,35,28)(24,39,40,48)(25,42,44,37)(27,34,45,32)(36,43,47,41);(9,11,13,12)(10,14,15,16)(17,20)(18,24)(19,27)(21,34)(22,31)(23,36)(25,43)(26,40)(28,46)(29,45)(30,33)(32,35)(37,48)(38,47)(39,42)(41,44);(1,3,4,7)(2,6,5,8)(20,31)(24,40)(27,45)(30,46)(32,34)(36,47)(39,48)(41,43);(2,5)(6,8)(9,12,13,11)(10,14,15,16)(19,28)(21,35)(23,37)(25,44)(27,46)(29,33)(30,45)(32,34)(36,48)(38,42)(39,47)(41,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,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, 6629606);