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 | |
4341 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)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,23,25)(19,24,28,30)(20,21,26,31)(22,32,29,27)(33,34,38,42)(35,43,45,39)(36,46,47,40)(37,41,44,48);(3,9)(7,13)(10,15)(14,16)(17,19)(18,24)(20,29)(21,27)(22,26)(23,28)(25,30)(31,32)(33,35,38,45)(34,39,42,43)(36,37,47,44)(40,41,46,48);(2,6)(7,13)(8,11)(14,16)(18,25)(19,28)(20,26)(27,32)(33,36,38,47)(34,40,42,46)(35,44,45,37)(39,48,43,41);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,23,26)(18,21,25,31)(19,29,28,22)(24,27,30,32)(33,36)(34,40)(35,37)(38,47)(39,41)(42,46)(43,48)(44,45);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,21)(18,26)(19,27)(20,25)(22,24)(23,31)(28,32)(29,30)(33,37)(34,41)(35,36)(38,44)(39,40)(42,48)(43,46)(45,47);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,22,23,29)(18,27,25,32)(19,26,28,20)(21,30,31,24)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,d^2=gi,f^2=gh,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, 9721777);