#8127443

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

4134 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:

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

Polycyclic presentation:

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

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, 8034639);