Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1822 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)(15,24)(16,28)(17,19)(18,22)(20,26)(21,32)(23,30)(25,41)(27,33)(29,40)(31,34)(35,39)(36,44)(37,43)(38,42);(2,4)(6,10)(11,12)(13,15,18,26)(14,20,22,24)(16,29,31,44)(17,33,23,39)(19,35,30,27)(21,37,38,41)(25,42,43,32)(28,36,34,40);(5,7,9,8)(6,11,10,12)(13,16,30,42,18,31,19,32)(14,21,17,34,22,38,23,28)(15,25,35,44,26,43,27,29)(20,36,39,41,24,40,33,37);(1,2,3,4)(13,17,30,22,18,23,19,14)(15,20,35,39,26,24,27,33)(16,28,42,21,31,34,32,38)(25,40,44,37,43,36,29,41);(5,8,9,7)(6,11,10,12)(14,22)(17,23)(20,24)(21,38)(28,34)(33,39)(36,40)(37,41)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^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521636);