Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| 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 | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2155 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)(8,9)(10,12)(13,14,18,22)(15,24,26,20)(16,27,31,44)(17,33,34,41)(19,35,29,23)(21,37,40,32)(25,42,43,38)(28,39,30,36);(1,2)(3,4)(6,9)(10,11)(13,15,18,26)(14,20,22,24)(16,28,31,30)(17,19,34,29)(21,38,40,42)(23,41,35,33)(25,37,43,32)(27,36,44,39);(13,16,29,37,18,31,19,32)(14,21,35,44,22,40,23,27)(15,25,34,30,26,43,17,28)(20,36,41,42,24,39,33,38);(2,4)(13,17,19,26,18,34,29,15)(14,20,23,33,22,24,35,41)(16,30,32,25,31,28,37,43)(21,39,27,42,40,36,44,38);(5,7,8,12)(6,10,9,11)(14,22)(16,31)(20,24)(23,35)(25,43)(28,30)(32,37)(33,41)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,d^2=g,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=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469794);