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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2203 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)(15,26)(16,30)(17,32)(18,23)(19,25)(20,24)(21,28)(22,35)(27,39)(29,38)(31,40)(33,37)(34,42)(36,41)(43,44);(1,2)(3,4)(6,10)(11,12)(13,15,18,28)(14,21,23,26)(16,19,29,20)(17,33,34,31)(22,39,41,44)(24,38,25,30)(27,35,43,36)(32,40,42,37);(5,7,9,8)(6,11,10,12)(13,16,19,28,18,29,20,15)(14,21,24,30,23,26,25,38)(17,33,35,27,34,31,36,43)(22,40,32,39,41,37,42,44);(2,4)(5,8,9,7)(6,12,10,11)(13,17,19,35,18,34,20,36)(14,22,24,32,23,41,25,42)(15,27,16,31,28,43,29,33)(21,37,30,44,26,40,38,39);(6,10)(11,12)(14,23)(17,34)(21,26)(24,25)(27,43)(30,38)(31,33)(35,36)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=fgi,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,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, 7469201);