#6840179

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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)(17,18,24,30)(19,25,33,41)(20,26,37,45)(21,27,23,29)(22,28,32,40)(31,39,47,48)(34,42,38,46)(35,43,36,44);(9,10)(11,14)(12,16)(13,15)(17,19,24,33)(18,25,30,41)(20,34,37,38)(21,32,23,22)(26,42,45,46)(27,40,29,28)(31,35,47,36)(39,43,48,44);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,20,21,35,24,37,23,36)(18,26,27,43,30,45,29,44)(19,31,22,38,33,47,32,34)(25,39,28,46,41,48,40,42);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,15,16)(17,21,24,23)(18,27,30,29)(19,32,33,22)(20,35,37,36)(25,40,41,28)(26,43,45,44)(31,34,47,38)(39,42,48,46);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,14,15,16)(17,22,23,19,24,32,21,33)(18,28,29,25,30,40,27,41)(20,34,36,47,37,38,35,31)(26,42,44,48,45,46,43,39)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fhi,d^2=gi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=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, 6747375);