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 | |
4286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,7,10,11)(5,8,13,12)(9,14,15,16)(17,18,22,24)(19,25,30,23)(20,26,27,21)(28,29,31,32)(35,41)(39,46)(42,44)(47,48);(2,4)(7,11)(8,12)(14,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,25)(26,32)(27,31)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47);(5,13)(8,12)(9,15)(14,16)(18,24)(21,26)(23,25)(29,32)(34,38)(39,46)(40,43)(47,48);(1,3,6,10)(2,7,4,11)(5,9,13,15)(8,14,12,16)(17,19)(18,25)(20,31)(21,29)(22,30)(23,24)(26,32)(27,28)(33,35)(34,39)(36,44)(37,41)(38,46)(40,48)(42,45)(43,47);(1,4,6,2)(3,7,10,11)(5,12,13,8)(9,14,15,16)(17,20,22,27)(18,26,24,21)(19,28,30,31)(23,32,25,29)(33,36)(34,40)(35,42)(37,45)(38,43)(39,47)(41,44)(46,48);(1,5,6,13)(2,8,4,12)(3,9,10,15)(7,14,11,16)(17,21,22,26)(18,27,24,20)(19,29,30,32)(23,31,25,28)(33,34)(35,39)(36,40)(37,38)(41,46)(42,47)(43,45)(44,48)Polycyclic presentation:
2^9|a^2=g,d^2=gh,e^2=g,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, 8832393);