#6729709

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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

1462 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)(19,23)(20,25)(21,24)(22,26)(27,36)(28,35)(29,37)(30,34)(31,38)(32,40)(33,39)(41,46)(42,45)(43,47)(44,48);(1,3)(2,5)(4,8)(6,7)(17,19,21,29)(18,23,24,37)(20,28,31,43)(22,27,33,30)(25,35,38,47)(26,36,39,34)(32,41,44,42)(40,46,48,45);(10,14)(15,16)(18,24)(19,27)(20,31)(22,33)(23,34)(28,42)(29,30)(35,46)(36,37)(40,48)(41,43)(45,47);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20,21,31)(18,25,24,38)(19,28,29,43)(22,32,33,44)(23,35,37,47)(26,40,39,48)(27,41,30,42)(34,45,36,46);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(19,27)(22,33)(23,36)(26,39)(28,41)(29,30)(32,44)(34,37)(35,46)(40,48)(42,43)(45,47)

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,h^c=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6636905);