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 | #52494 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4061 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)(3,8)(4,13)(5,9)(6,7)(10,14)(11,16)(12,15)(17,18,21,23)(19,25,26,22)(20,24,28,30)(27,31,32,29)(33,34,37,40)(35,38,42,47)(36,43,45,39)(41,48,44,46);(2,7)(8,14)(9,13)(15,16)(17,19)(18,22)(20,27)(21,26)(23,25)(24,29)(28,32)(30,31)(33,35)(34,38)(36,44)(37,42)(39,48)(40,47)(41,45)(43,46);(1,3,6,10)(2,8,7,14)(4,11,5,12)(9,15,13,16)(18,23)(22,25)(24,30)(29,31)(33,36,37,45)(34,39,40,43)(35,41,42,44)(38,46,47,48);(2,7)(3,10)(9,13)(11,12)(18,23)(19,26)(24,30)(27,32)(35,42)(38,47)(41,44)(46,48);(1,4,6,5)(2,9,7,13)(3,11,10,12)(8,15,14,16)(17,20,21,28)(18,24,23,30)(19,27,26,32)(22,29,25,31);(1,5,6,4)(2,9,7,13)(3,12,10,11)(8,15,14,16)Polycyclic presentation:
2^9|a^2=h,c^2=gh,e^2=g,f^2=ghi,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, 8000293);