#8726348

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

4198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8633544);