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 | |
1825 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)(15,23)(16,27)(17,32)(18,22)(19,34)(20,25)(21,31)(24,38)(26,37)(28,36)(29,30)(33,35)(39,43)(40,42)(41,44);(2,4)(6,10)(11,12)(13,15,18,25)(14,20,22,23)(16,28,30,42)(17,33,34,37)(19,35,32,26)(21,38,39,44)(24,31,41,43)(27,40,29,36);(5,7,9,8)(6,11,10,12)(13,16,19,31,18,30,32,43)(14,21,17,27,22,39,34,29)(15,24,26,42,25,41,35,28)(20,36,37,44,23,40,33,38);(1,2,3,4)(13,17,32,14,18,34,19,22)(15,23,35,37,25,20,26,33)(16,29,43,39,30,27,31,21)(24,36,28,38,41,40,42,44);(5,8,9,7)(6,11,10,12)(14,22)(17,34)(20,23)(21,39)(27,29)(33,37)(36,40)(38,44)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,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=cgi,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, 7521662);