Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #2673 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,10,5,7)(4,8,12,16)(11,15,13,14)(17,18,22,24)(19,27,28,23)(20,25,21,26)(29,32,30,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,47)(44,48);(1,3,6,5)(2,7,9,10)(4,11,12,13)(8,14,16,15)(17,19,22,28)(18,23,24,27)(20,29,21,30)(25,31,26,32)(33,35,37,42)(34,38,40,46)(36,43,44,41)(39,47,48,45);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,16)(10,15)(18,24)(23,27)(25,26)(31,32)(33,36)(34,39)(35,41)(37,44)(38,45)(40,48)(42,43)(46,47);(2,9)(4,12)(7,10)(11,13)(18,24)(19,28)(25,26)(29,30)(36,44)(39,48)(41,43)(45,47);(1,5,6,3)(2,10,9,7)(4,13,12,11)(8,15,16,14)(17,20,22,21)(18,25,24,26)(19,29,28,30)(23,31,27,32)(34,40)(38,46)(39,48)(45,47);(17,21,22,20)(18,26,24,25)(19,30,28,29)(23,32,27,31)Polycyclic presentation:
2^9|a^2=g,b^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,d^c=dghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999905);