Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806575 |
| 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×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1996 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)(6,9)(7,11)(15,24)(17,26)(20,28)(22,25);(5,6)(7,10)(8,9)(11,12)(13,14,18,23)(15,25,24,22)(16,21,19,27)(17,20,26,28);(1,2,3,4)(13,15,16,26,18,24,19,17)(14,20,27,22,23,28,21,25);(5,7,8,11)(6,10,9,12)(13,16,18,19)(14,21,23,27)(15,26,24,17)(20,25,28,22);(6,9)(10,12)(13,17,19,24,18,26,16,15)(14,22,21,20,23,25,27,28)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=fi,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722189);