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 | |
3998 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)(7,10)(11,15)(14,16)(17,18)(19,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32)(34,39)(37,45)(40,46)(44,48);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,18,23,20)(19,24,21,27)(22,28,30,25)(26,32,29,31)(36,43)(40,46)(42,47)(44,48);(2,6)(7,10)(8,12)(14,16)(17,19,23,21)(18,24,20,27)(22,26,30,29)(25,31,28,32)(33,34,38,39)(35,37,41,45)(36,40,43,46)(42,44,47,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,23,18)(19,24,21,27)(22,28,30,25)(26,31,29,32)(33,35,38,41)(34,37,39,45)(36,42,43,47)(40,44,46,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,21,23,19)(18,24,20,27)(22,29,30,26)(25,31,28,32)(33,36)(34,40)(35,42)(37,44)(38,43)(39,46)(41,47)(45,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22)(18,25)(19,26)(20,28)(21,29)(23,30)(24,31)(27,32)(33,37)(34,41)(35,39)(36,44)(38,45)(40,47)(42,46)(43,48)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8659303);