#6697133

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1588 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)(3,6)(4,5)(7,8)(10,14)(11,12)(17,18)(19,24)(20,26)(21,27)(22,25)(23,28)(29,43)(30,37)(31,38)(32,39)(33,41)(34,42)(35,40)(36,44)(45,47)(46,48);(9,10)(11,15)(12,16)(13,14)(17,19,22,32)(18,24,25,39)(20,30,23,29)(21,34,35,45)(26,37,28,43)(27,42,40,47)(31,36,46,33)(38,44,48,41);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(18,25)(19,29)(20,23)(21,35)(24,37)(30,32)(31,45)(34,46)(38,42)(39,43)(41,44)(47,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,20,22,23)(18,26,25,28)(19,30,32,29)(21,33,35,36)(24,37,39,43)(27,41,40,44)(31,45,46,34)(38,47,48,42);(10,14)(15,16)(17,21,22,35)(18,27,25,40)(19,31,32,46)(20,33,23,36)(24,38,39,48)(26,41,28,44)(29,34,30,45)(37,47,43,42)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,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, 6604329);