Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | |
2283 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,18,23)(15,25,29,20)(16,21,32,39)(17,22,19,24)(26,41,43,34)(27,36,28,35)(30,44,42,37)(31,38,33,40);(5,7)(6,9)(8,12)(10,11)(13,15,18,29)(14,20,23,25)(16,30,32,42)(17,28,19,27)(21,37,39,44)(22,36,24,35)(26,33,43,31)(34,40,41,38);(1,2,3,4)(13,16)(14,21)(15,26)(17,31)(18,32)(19,33)(20,34)(22,38)(23,39)(24,40)(25,41)(27,30)(28,42)(29,43)(35,37)(36,44);(6,10)(9,11)(13,17,18,19)(14,22,23,24)(15,27,29,28)(16,31,32,33)(20,35,25,36)(21,38,39,40)(26,30,43,42)(34,37,41,44);(7,12)(9,11)(15,28)(17,19)(20,36)(22,24)(25,35)(26,42)(27,29)(30,43)(31,33)(34,44)(37,41)(38,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589986);