#6840244

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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1356 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)(19,30)(20,26)(21,39)(22,40)(23,29)(24,27)(25,33)(28,32)(31,46)(34,44)(35,48)(36,42)(37,45)(38,43)(41,47);(9,10)(11,14)(12,16)(13,15)(17,19,23,33)(18,25,29,30)(20,34,37,36)(21,32,24,22)(26,42,45,44)(27,28,39,40)(31,35,47,38)(41,48,46,43);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,20,24,38,23,37,21,35)(18,26,27,43,29,45,39,48)(19,31,32,34,33,47,22,36)(25,41,40,42,30,46,28,44);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,15,16)(17,21,23,24)(18,27,29,39)(19,32,33,22)(20,35,37,38)(25,28,30,40)(26,43,45,48)(31,34,47,36)(41,44,46,42);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,14,15,16)(17,22,21,33,23,32,24,19)(18,28,39,30,29,40,27,25)(20,36,35,47,37,34,38,31)(26,44,48,46,45,42,43,41)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fhi,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^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, 6747440);