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 | #45385 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ 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)(7,9)(8,10)(11,12)(13,14,15,24,18,23,19,20)(16,27,28,21,30,34,25,37)(17,22,26,40,32,39,33,35)(29,43,42,36,31,44,41,38);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,18,19)(14,20,23,24)(16,28,30,25)(17,26,32,33)(21,27,37,34)(22,35,39,40)(29,42,31,41)(36,43,38,44);(5,7)(6,9)(8,12)(10,11)(13,16,18,30)(14,21,23,37)(15,25,19,28)(17,31,32,29)(20,34,24,27)(22,38,39,36)(26,42,33,41)(35,43,40,44);(1,3,4,2)(7,12)(9,11)(13,17,18,32)(14,22,23,39)(15,26,19,33)(16,29,30,31)(20,35,24,40)(21,36,37,38)(25,41,28,42)(27,43,34,44);(6,10)(9,11)(14,23)(17,32)(20,24)(21,37)(26,33)(27,34)(29,31)(41,42)Polycyclic presentation:
2^9|a^2=fg,b^2=i,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, 6913599);