#8260834

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

4060 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)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(34,40)(35,37)(36,44)(43,48);(1,2,6,5)(3,9,11,7)(4,12,13,8)(10,14,16,15)(18,22)(23,25)(24,29)(30,31)(33,34,39,40)(35,41,37,47)(36,45,44,38)(42,43,46,48);(2,5)(7,9)(8,12)(14,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35,39,37)(34,41,40,47)(36,42,44,46)(38,48,45,43);(1,3)(2,7)(4,10)(5,9)(6,11)(8,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36)(34,38)(35,42)(37,46)(39,44)(40,45)(41,48)(43,47);(1,4,6,13)(2,8,5,12)(3,10,11,16)(7,14,9,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,37,39,35)(34,41,40,47)(36,46,44,42)(38,48,45,43);(1,5,6,2)(3,9,11,7)(4,12,13,8)(10,15,16,14)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,38)(34,36)(35,43)(37,48)(39,45)(40,44)(41,46)(42,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=hi,e^2=gi,f^2=i,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, 8168030);