Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438949 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2016 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,28)(17,21)(18,23)(20,32)(22,30)(24,40)(25,34)(26,35)(27,39)(29,43)(31,38)(33,42)(36,41)(37,44);(2,3)(6,10)(11,12)(13,15)(14,19)(16,29)(17,27)(18,25)(20,36)(21,39)(22,35)(23,34)(24,31)(26,30)(28,43)(32,41)(33,37)(38,40)(42,44);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16,30,44,18,31,17,32)(14,20,21,38,23,37,22,28)(15,24,26,41,25,29,27,42)(19,33,39,43,34,36,35,40);(1,3,4,2)(5,8,9,7)(6,12,10,11)(15,25)(16,31)(19,34)(20,37)(26,27)(28,38)(32,44)(35,39);(13,17,18,30)(14,21,23,22)(15,26,25,27)(16,32,31,44)(19,35,34,39)(20,38,37,28)(24,41,29,42)(33,40,36,43)Polycyclic presentation:
2^9|a^2=hi,c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6364796);