Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2035 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,17,22,18,23,33,39)(15,24,28,36,27,19,26,37)(16,29,32,38,31,21,30,20)(25,35,42,34,43,41,44,40);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,18,27)(14,19,23,24)(16,25,31,43)(17,28,33,26)(20,34,38,40)(21,35,29,41)(22,37,39,36)(30,44,32,42);(5,7)(6,9)(8,12)(10,11)(13,16)(14,20)(15,25)(17,30)(18,31)(19,34)(21,22)(23,38)(24,40)(26,42)(27,43)(28,44)(29,39)(32,33)(35,37)(36,41);(1,3,4,2)(7,12)(9,11)(13,16)(14,21)(15,25)(17,32)(18,31)(19,35)(20,22)(23,29)(24,41)(26,44)(27,43)(28,42)(30,33)(34,37)(36,40)(38,39);(6,10)(9,11)(13,17,18,33)(14,22,23,39)(15,26,27,28)(16,30,31,32)(19,36,24,37)(20,21,38,29)(25,42,43,44)(34,41,40,35)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6913870);