Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1981 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)(7,11)(8,10)(9,12)(13,14,19,23)(15,25,26,20)(16,27,18,21)(17,24,28,22);(5,7)(6,9)(8,12)(10,11)(13,15,19,26)(14,20,23,25)(16,28,18,17)(21,22,27,24);(1,2,3,4)(13,16,17,15,19,18,28,26)(14,21,24,20,23,27,22,25);(6,10)(9,11)(13,17,19,28)(14,22,23,24)(15,16,26,18)(20,27,25,21);(7,12)(9,11)(13,18,28,15,19,16,17,26)(14,21,22,25,23,27,24,20)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=ghi,d^2=i,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722162);