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 | |
4268 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)(3,5)(4,11)(6,9)(7,8)(10,13)(12,16)(14,15)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,30)(31,32)(33,34)(35,44)(36,42)(37,48)(38,46)(39,43)(40,47)(41,45);(2,8)(5,11)(9,15)(13,16)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,32)(30,31)(33,35)(34,38)(36,37)(39,46)(40,48)(41,42)(43,44)(45,47);(1,3,7,4)(2,5,8,11)(6,10,14,12)(9,13,15,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,36)(34,40)(35,45)(37,46)(38,41)(39,47)(42,43)(44,48);(1,4,7,3)(2,5,8,11)(6,10,14,12)(9,16,15,13)(17,18)(19,20)(21,29)(22,25)(23,26)(24,30)(27,32)(28,31)(33,37,39,45)(34,41,43,48)(35,36,46,47)(38,42,44,40);(1,5)(2,4)(3,8)(6,13)(7,11)(9,12)(10,15)(14,16)(17,20)(18,23)(19,25)(21,28)(22,26)(24,31)(27,29)(30,32)(33,36)(34,42)(35,37)(38,41)(39,47)(40,43)(44,48)(45,46);(1,6)(2,9)(3,10)(4,12)(5,13)(7,14)(8,15)(11,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38)(34,35)(36,41)(37,42)(39,44)(40,45)(43,46)(47,48)Polycyclic presentation:
2^9|c^2=gh,d^2=g,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, 10315327);