Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14,19,25)(15,20,29,37)(16,22,31,39)(17,24,18,23)(21,32,38,30)(26,34,41,44)(27,36,28,35)(33,42,43,40);(1,2)(3,4)(6,9)(10,12)(13,15,19,29)(14,20,25,37)(16,26,31,41)(17,27,18,28)(21,33,38,43)(22,34,39,44)(23,35,24,36)(30,40,32,42);(2,4)(13,16)(14,21)(15,26)(17,30)(18,32)(19,31)(20,33)(22,23)(24,39)(25,38)(27,40)(28,42)(29,41)(34,35)(36,44)(37,43);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,22)(15,26)(17,32)(18,30)(19,31)(20,34)(21,23)(24,38)(25,39)(27,42)(28,40)(29,41)(33,35)(36,43)(37,44);(13,17,19,18)(14,23,25,24)(15,27,29,28)(16,30,31,32)(20,35,37,36)(21,22,38,39)(26,40,41,42)(33,34,43,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346537);