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×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2030 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)(7,10)(8,9)(11,12)(13,14,20,22)(15,21,25,23)(16,18,17,19)(24,27,28,26);(2,4)(6,9)(10,12)(13,15)(14,21)(16,26)(17,27)(18,24)(19,28)(20,25)(22,23);(5,7,8,11)(6,10,9,12)(13,16,19,14,20,17,18,22)(15,21,24,26,25,23,28,27);(1,2,3,4)(13,17,19,22,20,16,18,14)(15,23,24,27,25,21,28,26);(6,9)(10,12)(13,18,20,19)(14,17,22,16)(15,24,25,28)(21,27,23,26)Polycyclic presentation:
2^9|a^2=i,c^2=fg,d^2=h,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,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, 7520946);