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 | #23391 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2356 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)(7,12)(8,10)(9,11)(14,18)(19,27)(20,21)(22,28);(5,7,9,8)(6,10,11,12)(13,14,16,18)(15,22,23,28)(17,20,25,21)(19,26,27,24);(1,2,3,4)(13,15)(14,19)(16,23)(17,26)(18,27)(20,22)(21,28)(24,25);(5,8,9,7)(6,10,11,12)(14,20)(17,25)(18,21)(19,22)(24,26)(27,28);(15,23)(19,27)(22,28)(24,26)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=fi,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 | #72280745, #72310699, #72310705, … |
| maximal normal subgroup of | #72280745, #72310699, #72310705, … |
| maximal quotient of | #72312958, #72313001, #72313086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351005);