Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6636940 |
| 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 | #2615 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1445 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)(11,13)(12,14)(15,16)(17,18,22,24)(19,29,32,40)(20,34,35,26)(21,28,23,27)(25,31,38,33)(30,44,45,43)(36,41,37,42)(39,46,48,47);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(18,24)(20,35)(25,38)(27,28)(29,40)(30,45)(36,37)(46,47);(11,15)(13,16)(17,19,21,31,22,32,23,33)(18,25,27,40,24,38,28,29)(20,30,37,47,35,45,36,46)(26,39,42,44,34,48,41,43);(2,6)(3,8)(11,15)(13,16)(17,20,22,35)(18,26,24,34)(19,30,32,45)(21,37,23,36)(25,39,38,48)(27,42,28,41)(29,43,40,44)(31,47,33,46);(2,6)(5,7)(10,14)(13,16)(17,21,22,23)(18,27,24,28)(19,31,32,33)(20,36,35,37)(25,40,38,29)(26,41,34,42)(30,46,45,47)(39,43,48,44)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6544439);