Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438949 |
| 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×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,12)(8,10)(9,11)(13,14,18,22)(15,23,26,20)(16,25,27,24)(17,21,19,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,18,26)(14,20,22,23)(16,21,27,28)(17,24,19,25);(2,4)(13,16,19,20,18,27,17,23)(14,15,21,25,22,26,28,24);(2,4)(5,8,9,7)(6,10,11,12)(13,17,18,19)(14,21,22,28)(15,24,26,25)(16,20,27,23);(13,17,18,19)(14,21,22,28)(15,25,26,24)(16,23,27,20)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351088);