Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #42883 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2031 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)(7,9)(8,10)(11,12)(13,14,18,22,17,19,30,34)(15,23,26,32,24,38,31,20)(16,21,29,37,28,33,43,44)(25,39,40,36,27,41,42,35);(2,3)(5,6)(7,11)(8,10)(9,12)(14,19)(15,24)(21,33)(22,34)(25,27)(26,31)(37,44)(40,42);(5,7)(6,9)(8,12)(10,11)(13,15,17,24)(14,20,19,32)(16,27,28,25)(18,31,30,26)(21,36,33,35)(22,38,34,23)(29,40,43,42)(37,39,44,41);(1,3,4,2)(7,12)(9,11)(13,16,17,28)(14,21,19,33)(15,25,24,27)(18,29,30,43)(20,35,32,36)(22,37,34,44)(23,39,38,41)(26,40,31,42);(6,10)(9,11)(14,19)(16,28)(20,32)(22,34)(23,38)(25,27)(29,43)(40,42)Polycyclic presentation:
2^9|a^2=fgi,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6913607);