Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#469 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,15)(11,14,12,16)(18,23)(24,37)(25,31)(26,35)(27,39)(38,46)(40,41)(42,43);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,18,21,23)(19,27,29,39)(20,31,32,25)(22,35,34,26)(24,36,37,30)(28,43,44,42)(33,41,47,40)(38,45,46,48);(10,15)(14,16)(17,19,21,29)(18,24,23,37)(20,28,32,44)(22,36,34,30)(25,38,31,46)(26,39,35,27)(33,48,47,45)(40,42,41,43);(2,6)(5,7)(17,20)(18,25)(19,28)(21,32)(22,33)(23,31)(24,38)(26,41)(27,42)(29,44)(30,45)(34,47)(35,40)(36,48)(37,46)(39,43);(3,8)(5,7)(9,12,13,11)(10,16,15,14)(18,26)(20,32)(22,34)(23,35)(24,39)(25,40)(27,37)(28,44)(30,36)(31,41)(38,42)(43,46)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6625420);