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 | |
4259 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,7,10,11)(5,8,13,12)(9,14,15,16)(17,18,22,24)(19,23,25,20)(21,26,30,31)(27,29,32,28)(33,34)(35,39)(36,46)(37,41)(38,42)(40,44)(43,45)(47,48);(2,4)(7,11)(8,12)(14,16)(17,19,22,25)(18,23,24,20)(21,27,30,32)(26,29,31,28)(35,43)(37,47)(39,45)(41,48);(5,13)(8,12)(9,15)(14,16)(18,24)(19,25)(26,31)(27,32)(33,35,38,43)(34,39,42,45)(36,37,44,47)(40,48,46,41);(1,3,6,10)(2,7,4,11)(5,9,13,15)(8,14,12,16)(17,20,22,23)(18,25,24,19)(21,29,30,28)(26,27,31,32)(33,36)(34,40)(35,37)(38,44)(39,48)(41,45)(42,46)(43,47);(1,4,6,2)(3,7,10,11)(5,12,13,8)(9,14,15,16)(17,21,22,30)(18,26,24,31)(19,27,25,32)(20,28,23,29)(33,37)(34,41)(35,44)(36,43)(38,47)(39,40)(42,48)(45,46);(1,5,6,13)(2,8,4,12)(3,9,10,15)(7,14,11,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,29)(28,32)(30,31)(33,34)(35,45)(36,40)(37,41)(38,42)(39,43)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=hi,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833906);