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 | |
4228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,11,9)(4,12,13,8)(10,15,16,14)(17,18)(19,24)(20,22)(21,26)(23,25)(27,29)(28,31)(30,32)(33,34,38,40)(35,39,43,46)(36,37,44,45)(41,42,47,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,23,20)(18,24,25,22)(21,27,30,28)(26,29,32,31)(33,35,38,43)(34,39,40,46)(36,41,44,47)(37,42,45,48);(2,6)(7,9)(8,12)(14,15)(18,25)(19,20)(26,32)(27,28)(34,40)(36,44)(39,46)(41,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,23,19)(18,22,25,24)(21,28,30,27)(26,31,32,29)(33,36)(34,37)(35,41)(38,44)(39,42)(40,45)(43,47)(46,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,21)(18,26)(19,27)(20,28)(22,31)(23,30)(24,29)(25,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,22)(18,19)(20,25)(21,29)(23,24)(26,28)(27,32)(30,31)(33,37)(34,36)(35,42)(38,45)(39,41)(40,44)(43,48)(46,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9720170);