Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,9,11,12)(13,14,19,22)(15,25,26,23)(16,24,18,21)(17,20,27,28);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,15,16,27,19,26,18,17)(14,20,21,23,22,28,24,25);(6,10)(7,11)(13,16,19,18)(14,21,22,24)(15,17,26,27)(20,25,28,23);(2,4)(6,10)(9,12)(14,22)(15,26)(17,27)(21,24);(13,17,18,26,19,27,16,15)(14,23,24,20,22,25,21,28)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7434761);