Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,10)(7,11,12,9);(5,7,8,12)(6,9,10,11)(13,14,17,21)(15,22,23,28)(16,19,26,20)(18,25,27,24);(1,2,3,4)(13,15)(14,18)(16,25)(17,23)(19,22)(20,28)(21,27)(24,26);(14,19)(16,26)(18,22)(20,21)(24,25)(27,28);(15,23)(18,27)(22,28)(24,25)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72279952, #72280468, #72310637, … |
| maximal normal subgroup of | #72279952, #72280468, #72310637, … |
| maximal quotient of | #72313054, #72313237, #72313336, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350601);