#6840288

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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1370 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)(19,30,32,25)(20,26,36,45)(21,38,24,27)(22,39,40,28)(31,46,47,41)(33,44,35,42)(34,48,37,43);(9,10)(11,14)(12,16)(13,15)(17,19,23,32)(18,25,29,30)(20,33,36,35)(21,22,24,40)(26,42,45,44)(27,39,38,28)(31,34,47,37)(41,48,46,43);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,20,21,34,23,36,24,37)(18,26,38,48,29,45,27,43)(19,31,40,35,32,47,22,33)(25,41,39,44,30,46,28,42);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,15,16)(17,21,23,24)(18,27,29,38)(19,22,32,40)(20,34,36,37)(25,39,30,28)(26,43,45,48)(31,33,47,35)(41,44,46,42);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,14,15,16)(17,22,21,19,23,40,24,32)(18,28,38,25,29,39,27,30)(20,35,34,47,36,33,37,31)(26,44,48,46,45,42,43,41)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fh,d^2=gi,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,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, 6747484);