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 |
| prime radical | #59139 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1827 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,9)(7,11,12,10)(13,14,18,22)(15,24,16,20)(17,27,26,21)(19,28,25,23);(1,2)(3,4)(6,9)(10,11)(13,15,18,16)(14,20,22,24)(17,19,26,25)(21,28,27,23);(13,16,19,26,18,15,25,17)(14,21,23,20,22,27,28,24);(2,4)(13,17,19,16,18,26,25,15)(14,20,23,27,22,24,28,21);(5,7,8,12)(6,10,9,11)(14,22)(20,24)(21,27)(23,28)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=g,e^2=fi,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,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72308484, #72308528 |
| maximal normal subgroup of | #72308484, #72308528 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470028);