#8260844

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4067 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,d^2=g,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8168040);