#6729773

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1457 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=g,e^2=gi,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, 6636969);