#6840178

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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1374 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)(17,18,23,29)(19,25,33,42)(20,34,37,26)(21,27,24,30)(22,28,32,41)(31,47,48,40)(35,46,39,43)(36,45,38,44);(9,10)(11,14)(12,16)(13,15)(17,19,23,33)(18,25,29,42)(20,35,37,39)(21,32,24,22)(26,43,34,46)(27,41,30,28)(31,36,48,38)(40,44,47,45);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,20,21,36,23,37,24,38)(18,26,27,44,29,34,30,45)(19,31,22,39,33,48,32,35)(25,40,28,46,42,47,41,43);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,15,16)(17,21,23,24)(18,27,29,30)(19,32,33,22)(20,36,37,38)(25,41,42,28)(26,44,34,45)(31,35,48,39)(40,43,47,46);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,14,15,16)(17,22,24,19,23,32,21,33)(18,28,30,25,29,41,27,42)(20,35,38,48,37,39,36,31)(26,43,45,47,34,46,44,40)

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,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, 6747374);