Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500186 |
| 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×#3198 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2100 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)(13,14,18,22,17,23,16,26)(15,21,24,28,25,19,27,20);(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,15)(14,19)(16,27)(17,25)(18,24)(20,22)(21,23)(26,28);(5,7)(6,9)(8,12)(10,11)(13,15,17,25)(14,20,23,28)(16,24,18,27)(19,26,21,22);(7,12)(9,11)(13,15)(14,21)(16,27)(17,25)(18,24)(19,23)(20,26)(22,28);(6,10)(9,11)(13,16,17,18)(14,22,23,26)(15,24,25,27)(19,28,21,20)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6410156);