Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470428 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,25)(19,28,32,42)(20,26,33,39)(21,27,22,36)(24,30,37,31)(29,43,44,48)(34,40,35,41)(38,46,47,45);(9,11)(10,13)(12,15)(14,16)(17,19,21,30,23,32,22,31)(18,24,36,42,25,37,27,28)(20,29,35,46,33,44,34,45)(26,38,40,48,39,47,41,43);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(18,25)(20,33)(24,37)(27,36)(28,42)(29,44)(34,35)(45,46);(11,15)(13,16)(17,20)(18,26)(19,29)(21,35)(22,34)(23,33)(24,38)(25,39)(27,41)(28,43)(30,46)(31,45)(32,44)(36,40)(37,47)(42,48);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,21,23,22)(18,27,25,36)(19,30,32,31)(20,34,33,35)(24,28,37,42)(26,40,39,41)(29,45,44,46)(38,48,47,43)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6384258);