Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500076 |
| 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×#3188 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2204 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)(13,14,18,22,17,21,16,23)(15,19,25,20,24,28,26,27);(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,15,17,24)(14,19,21,28)(16,26,18,25)(20,23,27,22);(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,25)(17,24)(18,26)(19,22)(21,27)(23,28);(7,12)(9,11)(14,21)(15,24)(22,23)(25,26);(6,10)(9,11)(13,16,17,18)(14,22,21,23)(15,25,24,26)(19,27,28,20)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6409926);