Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| 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 | #24244 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1845 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,12)(9,10)(13,14)(15,21)(16,30)(17,32)(18,24)(19,25)(20,26)(22,29)(23,34)(27,43)(28,38)(31,37)(33,39)(35,42)(36,41)(40,44);(2,4)(6,10)(11,12)(13,15,20,29)(14,21,26,22)(16,18,28,19)(17,33,36,31)(23,40,42,43)(24,38,25,30)(27,34,44,35)(32,39,41,37);(5,7,9,8)(6,11,10,12)(13,16,18,29,20,28,19,15)(14,22,25,30,26,21,24,38)(17,33,34,27,36,31,35,44)(23,37,32,43,42,39,41,40);(1,2,3,4)(13,17,18,34,20,36,19,35)(14,23,25,32,26,42,24,41)(15,27,16,31,29,44,28,33)(21,37,38,43,22,39,30,40);(5,8,9,7)(6,11,10,12)(13,18,20,19)(14,24,26,25)(15,28,29,16)(17,34,36,35)(21,38,22,30)(23,41,42,32)(27,33,44,31)(37,43,39,40)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=hi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521706);