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 | |
4142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,8,10,4)(5,9,13,14)(11,15,16,12)(19,25)(23,26)(27,31)(30,32)(33,34)(35,37)(36,41)(38,40)(39,42)(43,46)(44,45)(47,48);(2,7)(4,8)(9,14)(12,15)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32)(33,35,38,42)(34,39,40,37)(36,43,44,48)(41,47,45,46);(3,10)(4,8)(5,13)(9,14)(18,22)(19,25)(20,28)(30,32)(34,40)(36,44)(37,39)(43,48);(1,3,6,10)(2,8,7,4)(5,11,13,16)(9,15,14,12)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,34)(35,39)(36,45)(37,42)(38,40)(41,44)(43,46)(47,48);(1,4,6,8)(2,3,7,10)(5,12,13,15)(9,11,14,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,36,38,44)(34,41,40,45)(35,43,42,48)(37,46,39,47);(1,5,6,13)(2,9,7,14)(3,11,10,16)(4,12,8,15)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,37,38,39)(34,42,40,35)(36,46,44,47)(41,48,45,43)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8964553);