Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628847 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4004 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)(13,14,18,22)(15,19,24,33)(16,26,27,36)(17,29,31,21)(20,30,35,28)(23,37,38,44)(25,41,40,34)(32,42,43,39);(6,9)(10,12)(13,15)(14,19)(16,23)(17,25)(18,24)(20,32)(21,34)(22,33)(26,37)(27,38)(28,39)(29,41)(30,42)(31,40)(35,43)(36,44);(13,16,18,27)(14,20,22,35)(15,23,24,38)(17,30,31,28)(19,32,33,43)(21,36,29,26)(25,42,40,39)(34,44,41,37);(2,4)(15,24)(19,33)(23,38)(25,40)(32,43)(34,41)(37,44)(39,42);(5,7,8,11)(6,10,9,12)(16,27)(20,35)(23,38)(26,36)(28,30)(32,43)(37,44)(39,42)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,d^b=di,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536532);