#8581454

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

4081 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8488650);