#6730754

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	#30118	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1491 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)(18,23)(20,31)(24,37)(26,35)(27,40)(28,32)(33,45)(44,47);(1,3,4,8)(2,5,6,7)(17,18)(19,27)(20,25)(21,23)(22,35)(24,30)(26,34)(28,43)(29,40)(31,38)(32,46)(33,48)(36,37)(39,44)(41,47)(42,45);(10,14)(15,16)(17,19,21,29)(18,24,23,37)(20,32,31,28)(22,36,34,30)(25,41,38,39)(26,40,35,27)(33,44,45,47)(42,43,48,46);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20,21,31)(18,25,23,38)(19,28,29,32)(22,33,34,45)(24,39,37,41)(26,42,35,48)(27,43,40,46)(30,44,36,47);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(18,26)(22,34)(23,35)(24,40)(25,42)(27,37)(30,36)(33,45)(38,48)(39,46)(41,43)(44,47)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^2=gi,e^2=g,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, 6637950);