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 | |
4097 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,12)(3,9,4,8)(5,6,11,10)(13,14,15,16)(17,18,24,22)(19,28,23,25)(20,27,21,26)(29,31,30,32)(33,34,39,43)(35,40,44,48)(36,38,42,47)(37,46,45,41);(1,3,7,4)(2,8,12,9)(5,15,11,13)(6,14,10,16)(17,19,24,23)(18,25,22,28)(20,29,21,30)(26,31,27,32)(35,44)(37,45)(40,48)(41,46);(17,20,24,21)(18,26,22,27)(19,29,23,30)(25,31,28,32)(33,35,39,44)(34,40,43,48)(36,45,42,37)(38,41,47,46);(1,4,7,3)(2,9,12,8)(5,15,11,13)(6,14,10,16)(17,21,24,20)(18,27,22,26)(19,29,23,30)(25,31,28,32)(33,36)(34,38)(35,45)(37,44)(39,42)(40,41)(43,47)(46,48);(1,5,7,11)(2,10,12,6)(3,13,4,15)(8,14,9,16)(17,22,24,18)(19,25,23,28)(20,27,21,26)(29,31,30,32)(33,37,39,45)(34,41,43,46)(35,36,44,42)(38,40,47,48);(1,6,7,10)(2,11,12,5)(3,14,4,16)(8,13,9,15)(17,23,24,19)(18,25,22,28)(20,30,21,29)(26,31,27,32)(33,38)(34,42)(35,41)(36,43)(37,40)(39,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=hi,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8399990);