Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499804 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1816 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(13,14)(15,20)(16,28)(17,22)(18,31)(19,23)(21,30)(24,38)(25,34)(26,35)(27,41)(29,37)(32,36)(33,39)(40,43)(42,44);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15,19,25)(14,20,23,34)(16,24,29,40)(17,26,18,27)(21,33,36,44)(22,35,31,41)(28,38,37,43)(30,39,32,42);(1,3,4,2)(6,9)(7,11)(13,16,19,29)(14,21,23,36)(15,24,25,40)(17,30,18,32)(20,33,34,44)(22,37,31,28)(26,39,27,42)(35,43,41,38);(5,7,8,11)(6,10,9,12)(15,25)(16,29)(20,34)(21,36)(26,27)(28,37)(30,32)(35,41);(6,9)(10,12)(13,17,19,18)(14,22,23,31)(15,26,25,27)(16,30,29,32)(20,35,34,41)(21,37,36,28)(24,39,40,42)(33,43,44,38)Polycyclic presentation:
2^9|b^2=fi,c^2=fi,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6421373);