Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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 | C2×#3269 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2013 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(13,14,19,32,18,22,17,23)(15,20,26,34,25,36,27,35)(16,28,29,21,31,38,30,37)(24,39,42,33,40,44,41,43);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,25)(14,20,22,36)(16,24,31,40)(17,27,19,26)(21,33,37,43)(23,35,32,34)(28,39,38,44)(29,42,30,41);(6,9)(10,12)(13,16,18,31)(14,21,22,37)(15,24,25,40)(17,29,19,30)(20,33,36,43)(23,38,32,28)(26,41,27,42)(34,39,35,44);(5,7)(6,10)(8,11)(9,12)(14,22)(15,25)(21,37)(23,32)(24,40)(26,27)(28,38)(41,42);(7,11)(10,12)(13,17,18,19)(14,23,22,32)(15,26,25,27)(16,29,31,30)(20,34,36,35)(21,38,37,28)(24,41,40,42)(33,39,43,44)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ei,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6452326);