#6840530

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1372 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)(3,7)(4,8)(5,6)(9,10,13,15)(11,16,12,14)(17,18)(19,29)(20,25)(21,36)(22,26)(23,28)(24,31)(27,32)(30,45)(33,44)(34,41)(35,42)(37,43)(38,40)(39,47)(46,48);(9,10)(11,14)(12,16)(13,15)(17,19,22,31)(18,24,26,29)(20,33,34,38)(21,37,32,23)(25,40,41,44)(27,43,36,28)(30,46,47,35)(39,48,45,42);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,20,23,35,22,34,37,46)(18,25,28,42,26,41,43,48)(19,30,32,33,31,47,21,38)(24,39,36,40,29,45,27,44);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,15,16)(18,26)(20,34)(24,29)(27,36)(28,43)(30,47)(33,38)(35,46);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,14,15,16)(17,21,37,31,22,32,23,19)(18,27,43,29,26,36,28,24)(20,33,46,30,34,38,35,47)(25,40,48,39,41,44,42,45)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fhi,d^2=g,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei

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, 6747726);