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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,10)(7,11,12,9)(13,14,18,24)(15,25,26,20)(16,22,28,21)(17,27,19,23);(1,2,3,4)(13,15,18,26)(14,20,24,25)(16,27,28,23)(17,21,19,22);(5,7,8,12)(6,9,10,11)(13,16,17,25,18,28,19,20)(14,21,27,15,24,22,23,26);(13,16,19,20,18,28,17,25)(14,22,23,15,24,21,27,26);(13,17,18,19)(14,23,24,27)(15,21,26,22)(16,25,28,20)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=gh,d^2=h,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373440);