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 | #59131 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,10)(7,11,8,12)(13,14,17,21)(15,23,24,19)(16,20,25,28)(18,27,26,22);(1,2)(3,4)(6,10)(11,12)(13,15,17,24)(14,19,21,23)(16,18,25,26)(20,22,28,27);(5,7,9,8)(6,11,10,12)(13,15,18,16,17,24,26,25)(14,20,22,23,21,28,27,19);(2,4)(5,7,9,8)(6,11,10,12)(13,16,18,24,17,25,26,15)(14,19,22,20,21,23,27,28);(5,8,9,7)(6,11,10,12)(14,21)(19,23)(20,28)(22,27)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgi,d^2=fgi,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 | #72308261, #72308528 |
| maximal normal subgroup of | #72308261, #72308528 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7472957);