Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #45332 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,24,20,25,18,35)(15,22,27,37,16,28,30,21)(17,23,33,39,34,40,32,44)(26,38,31,43,29,41,42,36);(5,6)(7,11)(8,10)(9,12)(13,15,20,16)(14,21,25,37)(17,31,34,42)(18,27,19,30)(22,24,28,35)(23,38,40,41)(26,32,29,33)(36,44,43,39);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,16,20,15)(14,22,25,28)(17,29,34,26)(18,27,19,30)(21,24,37,35)(23,38,40,41)(31,33,42,32)(36,39,43,44);(7,12)(9,11)(13,17,18,32,20,34,19,33)(14,23,35,44,25,40,24,39)(15,26,30,42,16,29,27,31)(21,36,28,41,37,43,22,38);(6,10)(9,11)(13,18,20,19)(14,24,25,35)(15,27,16,30)(17,32,34,33)(21,28,37,22)(23,39,40,44)(26,31,29,42)(36,41,43,38)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6928790);