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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,11)(8,9)(10,12)(13,14,18,22)(15,24,26,20)(16,27,30,43)(17,32,33,39)(19,34,29,23)(21,36,38,31)(25,41,42,37)(28,40,44,35);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15,18,26)(14,20,22,24)(16,28,30,44)(17,19,33,29)(21,37,38,41)(23,39,34,32)(25,36,42,31)(27,35,43,40);(6,9)(10,11)(13,16,29,36,18,30,19,31)(14,21,34,43,22,38,23,27)(15,25,33,44,26,42,17,28)(20,35,39,41,24,40,32,37);(5,7,8,12)(6,10,9,11)(13,17,19,26,18,33,29,15)(14,20,23,32,22,24,34,39)(16,28,31,42,30,44,36,25)(21,35,27,37,38,40,43,41);(1,3,4,2)(14,22)(16,30)(20,24)(23,34)(25,42)(28,44)(31,36)(32,39)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=hi,d^2=g,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,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504280);