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 | |
4216 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)(3,4)(5,13)(6,7)(8,15)(9,12)(10,14)(11,16)(17,18,23,25)(19,28,22,24)(20,21,27,26)(29,30,31,32)(33,34,38,41)(35,37,43,39)(36,40,44,47)(42,45,48,46);(1,3,6,9)(2,4,7,12)(5,10,15,16)(8,11,13,14)(17,19,23,22)(18,24,25,28)(20,29,27,31)(21,32,26,30)(36,44)(40,47)(42,48)(45,46);(2,7)(3,9)(8,13)(10,16)(18,25)(20,27)(24,28)(29,31)(33,35,38,43)(34,39,41,37)(36,42,44,48)(40,46,47,45);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(13,15)(14,16)(17,20,23,27)(18,26,25,21)(19,29,22,31)(24,30,28,32)(33,35)(34,37)(36,42)(38,43)(39,41)(40,45)(44,48)(46,47);(1,4)(2,3)(5,14)(6,12)(7,9)(8,16)(10,13)(11,15)(17,21,23,26)(18,27,25,20)(19,30,22,32)(24,29,28,31)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,5,6,15)(2,8,7,13)(3,10,9,16)(4,11,12,14)(17,22,23,19)(18,28,25,24)(20,31,27,29)(21,30,26,32)(33,37)(34,35)(36,45)(38,39)(40,42)(41,43)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 9619422);