Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,7)(4,5)(6,8)(9,13)(10,11)(12,14)(15,16)(17,18,22,23)(19,24,27,30)(20,25,31,28)(21,29,32,26)(33,34,39,37)(35,43,44,40)(36,45,46,41)(38,42,48,47);(1,3,6,12)(2,7,8,14)(4,10,9,15)(5,11,13,16)(17,18,22,23)(19,24,27,30)(20,28,31,25)(21,26,32,29)(33,35)(34,40)(36,38)(37,43)(39,44)(41,42)(45,47)(46,48);(2,8)(4,9)(7,14)(10,15)(18,23)(19,27)(25,28)(26,29)(33,34,39,37)(35,40,44,43)(36,41,46,45)(38,42,48,47);(1,4,6,9)(2,5,8,13)(3,10,12,15)(7,11,14,16)(17,19)(18,24)(20,29)(21,28)(22,27)(23,30)(25,32)(26,31)(33,36)(34,41)(35,38)(37,45)(39,46)(40,42)(43,47)(44,48);(1,3)(2,7)(4,10)(5,11)(6,12)(8,14)(9,15)(13,16)(17,20)(18,25)(19,26)(21,24)(22,31)(23,28)(27,29)(30,32)(33,37,39,34)(35,43,44,40)(36,41,46,45)(38,42,48,47);(1,5)(2,9)(3,11)(4,8)(6,13)(7,15)(10,14)(12,16)(17,21)(18,26)(19,28)(20,30)(22,32)(23,29)(24,31)(25,27)(33,38)(34,42)(35,36)(37,47)(39,48)(40,41)(43,45)(44,46)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=g,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10028440);