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 | |
4254 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)(6,8)(7,11)(9,15)(10,16)(12,14)(17,18)(19,24)(20,23)(21,25)(22,26)(27,30)(28,31)(29,32)(34,39)(35,42)(37,45)(44,47);(1,3,6,11)(2,7,8,4)(5,10,15,14)(9,16,13,12)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,32,28,30)(33,34,38,39)(35,36,42,40)(37,41,45,46)(43,44,48,47);(2,8)(4,7)(9,13)(12,16)(18,20)(19,26)(25,28)(27,29)(33,35,38,42)(34,40,39,36)(37,43,45,48)(41,47,46,44);(1,4)(2,3)(5,12)(6,7)(8,11)(9,10)(13,14)(15,16)(17,20,23,18)(19,22,26,24)(21,28,31,25)(27,32,29,30)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,3)(2,4)(5,14)(6,11)(7,8)(9,16)(10,15)(12,13)(17,21,23,31)(18,25,20,28)(19,27,26,29)(22,32,24,30)(33,36)(34,35)(37,44)(38,40)(39,42)(41,43)(45,47)(46,48);(1,5)(2,9)(3,10)(4,12)(6,15)(7,16)(8,13)(11,14)(17,22)(18,19)(20,26)(21,30)(23,24)(25,29)(27,28)(31,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|b^2=h,c^2=ghi,d^2=gh,e^2=gh,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, 9633820);