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 | |
4260 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)(19,20)(21,29)(22,23)(24,26)(25,31)(27,30)(28,32)(33,34,38,39)(35,41,46,40)(36,42,47,45)(37,44,48,43);(3,9)(7,13)(10,15)(14,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,30)(31,32)(34,39)(36,47)(40,41)(43,44);(2,6)(7,13)(8,11)(14,16)(18,23)(19,26)(25,29)(27,32)(33,35,38,46)(34,40,39,41)(36,44,47,43)(37,45,48,42);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19)(18,24)(20,23)(21,27)(22,26)(25,30)(28,29)(31,32)(33,35)(34,41)(36,44)(37,45)(38,46)(39,40)(42,48)(43,47);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20,22,24)(18,19,23,26)(21,30,31,28)(25,32,29,27)(33,36)(34,42)(35,44)(37,40)(38,47)(39,45)(41,48)(43,46);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,31)(18,25,23,29)(19,27,26,32)(20,28,24,30)(33,37,38,48)(34,43,39,44)(35,45,46,42)(36,41,47,40)Polycyclic presentation:
2^9|a^2=g,c^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,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, 9621965);