Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | #42926 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2213 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,27)(16,22)(17,23)(18,24)(19,25)(20,26)(21,31)(28,43)(29,44)(30,40)(32,42)(33,39)(34,38)(35,37)(36,41);(5,7)(6,9)(8,12)(10,11)(13,15,19,31)(14,21,25,27)(16,32,30,34)(17,29,20,35)(18,33,28,36)(22,38,40,42)(23,37,26,44)(24,41,43,39);(1,2,3,4)(13,16,19,30)(14,22,25,40)(15,28,31,18)(17,33,20,36)(21,24,27,43)(23,39,26,41)(29,34,35,32)(37,42,44,38);(6,10)(9,11)(13,17,19,20)(14,23,25,26)(15,29,31,35)(16,33,30,36)(18,32,28,34)(21,37,27,44)(22,39,40,41)(24,42,43,38);(7,12)(9,11)(13,18,19,28)(14,24,25,43)(15,30,31,16)(17,34,20,32)(21,22,27,40)(23,38,26,42)(29,33,35,36)(37,41,44,39)Polycyclic presentation:
2^9|b^2=i,c^2=gi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,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, 6589911);