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 | |
3986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,14)(10,12)(15,16)(17,18,22,24)(19,23,25,20)(21,26,30,31)(27,29,32,28)(33,34)(35,42)(36,37)(38,39)(40,45)(41,46)(43,48)(44,47);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,19,22,25)(18,23,24,20)(21,27,30,32)(26,29,31,28)(34,39)(36,41)(40,42)(43,47);(2,6)(7,14)(8,11)(15,16)(18,24)(19,25)(26,31)(27,32)(33,34,38,39)(35,40,45,42)(36,37,41,46)(43,44,47,48);(1,3)(2,7)(4,12)(5,9)(6,14)(8,16)(10,13)(11,15)(17,20,22,23)(18,25,24,19)(21,29,30,28)(26,27,31,32)(33,35,38,45)(34,40,39,42)(36,43,41,47)(37,44,46,48);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)(33,36,38,41)(34,37,39,46)(35,43,45,47)(40,44,42,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,29)(28,32)(30,31)(33,37)(34,41)(35,44)(36,39)(38,46)(40,47)(42,43)(45,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,d^2=h,e^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, 8832512);