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×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2128 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)(7,11)(8,9)(10,12)(13,14,19,23)(15,22,24,28)(16,21,18,27)(17,26,25,20);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,25,19,24,16,17)(14,20,21,22,23,26,27,28);(6,9)(10,11)(13,16,19,18)(14,21,23,27)(15,17,24,25)(20,22,26,28);(2,4)(5,7,8,12)(6,10,9,11)(15,24)(17,25)(20,26)(22,28);(13,17,16,24,19,25,18,15)(14,22,27,20,23,28,21,26)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=g,e^2=fi,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^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7458328);