Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499814 |
| 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 | C2×#3005 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2760 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)(13,14,17,21)(15,24,25,19)(16,23,18,22)(20,27,28,26);(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,15,17,25)(14,19,21,24)(16,27,18,26)(20,22,28,23);(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,26)(17,25)(18,27)(19,23)(21,28)(22,24);(7,12)(9,11)(14,21)(15,25)(22,23)(26,27);(6,10)(9,11)(13,16,17,18)(14,22,21,23)(15,26,25,27)(19,20,24,28)Polycyclic presentation:
2^9|a^2=i,b^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407563);