Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437415 |
| 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 | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3107 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)(7,10)(8,9)(11,12)(13,14,18,21)(15,23,26,36)(16,20,27,34)(17,29,31,22)(19,30,33,25)(24,37,38,44)(28,41,40,35)(32,42,43,39);(1,2)(3,4)(13,15,18,26)(14,19,21,33)(16,24,27,38)(17,30,31,25)(20,32,34,43)(22,36,29,23)(28,42,40,39)(35,44,41,37);(2,4)(6,9)(10,12)(13,16,18,27)(14,20,21,34)(15,24,26,38)(17,28,31,40)(19,32,33,43)(22,35,29,41)(23,37,36,44)(25,39,30,42);(2,4)(16,27)(20,34)(24,38)(28,40)(32,43)(35,41)(37,44)(39,42);(5,7,8,11)(6,10,9,12)(14,21)(19,33)(20,34)(22,29)(23,36)(32,43)(35,41)(37,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6360946);