Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628851 |
| 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 | |
4010 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)(5,6,8,10)(7,11,12,9)(13,14,17,19)(15,23,25,36)(16,21,27,33)(18,30,29,22)(20,31,32,26)(24,37,38,44)(28,41,40,35)(34,42,43,39);(5,7,8,12)(6,9,10,11)(14,19)(20,32)(21,33)(22,30)(23,36)(34,43)(35,41)(37,44);(13,15,17,25)(14,20,19,32)(16,24,27,38)(18,31,29,26)(21,34,33,43)(22,36,30,23)(28,42,40,39)(35,44,41,37);(2,4)(13,16)(14,21)(15,24)(17,27)(18,28)(19,33)(20,34)(22,35)(23,37)(25,38)(26,39)(29,40)(30,41)(31,42)(32,43)(36,44);(16,27)(21,33)(24,38)(28,40)(34,43)(35,41)(37,44)(39,42)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,g^c=gi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536288);