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 | |
4045 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,6,11,9)(4,7,13,16)(10,14,12,15)(18,23)(19,26)(21,29)(28,31)(33,34,39,44)(35,42,46,40)(36,47,43,38)(37,45,48,41);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,18,22,23)(19,20,26,24)(21,25,29,30)(27,28,32,31)(33,35,39,46)(34,40,44,42)(36,37,43,48)(38,45,47,41);(17,19,22,26)(18,24,23,20)(21,27,29,32)(25,31,30,28)(33,36,39,43)(34,38,44,47)(35,37,46,48)(40,45,42,41);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,32)(30,31)(33,37,39,48)(34,41,44,45)(35,43,46,36)(38,42,47,40);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20)(18,19)(21,28)(22,24)(23,26)(25,27)(29,31)(30,32)(33,35)(34,42)(36,37)(38,41)(39,46)(40,44)(43,48)(45,47);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,21,22,29)(18,25,23,30)(19,27,26,32)(20,28,24,31)(33,38)(34,43)(35,45)(36,44)(37,42)(39,47)(40,48)(41,46)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=h,d^2=hi,e^2=gh,f^2=ghi,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, 10109289);