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 | |
3969 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,4,11,7)(5,8,14,16)(10,12,15,13)(17,18,21,25)(19,23,27,22)(20,28,29,24)(26,30,32,31)(34,39)(37,40)(41,47)(45,48);(1,3,6,11)(2,7,9,4)(5,10,14,15)(8,13,16,12)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32);(17,18)(19,23)(20,24)(21,25)(22,27)(26,31)(28,29)(30,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,48)(44,45)(46,47);(1,4)(2,3)(5,13)(6,7)(8,15)(9,11)(10,16)(12,14)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,35,38,42)(34,40,39,37)(36,44,46,43)(41,45,47,48);(1,3)(2,7)(4,9)(5,10)(6,11)(8,13)(12,16)(14,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,36,38,46)(34,41,39,47)(35,43,42,44)(37,45,40,48);(1,5,6,14)(2,8,9,16)(3,10,11,15)(4,12,7,13)(17,20,21,29)(18,24,25,28)(19,26,27,32)(22,30,23,31)(33,37,38,40)(34,42,39,35)(36,45,46,48)(41,44,47,43)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=hi,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10358846);