Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431710 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,22)(17,23)(18,24)(19,25)(21,32)(26,42)(27,35)(28,36)(29,37)(30,38)(31,41)(33,40)(34,43)(39,44);(6,9)(10,12)(13,15,18,29)(14,20,24,37)(16,27,32,44)(17,30,19,28)(21,39,22,35)(23,38,25,36)(26,33,43,31)(34,41,42,40);(2,4)(13,16)(14,21)(15,26)(17,31)(18,32)(19,33)(20,34)(22,24)(23,40)(25,41)(27,28)(29,43)(30,44)(35,38)(36,39)(37,42);(5,7,8,11)(6,10,9,12)(13,16)(14,22)(15,27)(17,33)(18,32)(19,31)(20,35)(21,24)(23,40)(25,41)(26,28)(29,44)(30,43)(34,38)(36,42)(37,39);(13,17,18,19)(14,23,24,25)(15,28,29,30)(16,31,32,33)(20,36,37,38)(21,40,22,41)(26,27,43,44)(34,39,42,35)Polycyclic presentation:
2^9|a^2=gi,b^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ei,h^b=hi,h^c=hi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339575);