Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| 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 | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2634 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)(13,14,18,22)(15,23,26,36)(16,27,28,20)(17,30,32,21)(19,31,34,25)(24,38,39,37)(29,35,43,42)(33,40,41,44);(1,2,3,4)(5,6)(7,11)(8,10)(9,12);(5,7)(6,9)(8,12)(10,11)(13,15,18,26)(14,19,22,34)(16,24,28,39)(17,31,32,25)(20,33,27,41)(21,36,30,23)(29,44,43,40)(35,38,42,37);(7,12)(9,11)(13,16,18,28)(14,20,22,27)(15,24,26,39)(17,29,32,43)(19,33,34,41)(21,35,30,42)(23,37,36,38)(25,40,31,44);(6,10)(9,11)(16,28)(20,27)(24,39)(29,43)(33,41)(35,42)(37,38)(40,44)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407725);