Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #24362 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2021 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(7,11)(10,12)(13,14)(15,22)(16,20)(17,21)(18,19)(23,27)(24,26)(25,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,18)(14,19,21,22)(16,24,25,23)(20,27,28,26);(6,9)(10,12)(14,19)(15,18)(20,26)(21,22)(23,24)(27,28);(6,9)(10,12)(13,16,17,25)(14,20,21,28)(15,23,18,24)(19,26,22,27);(5,7,8,11)(6,10,9,12)(14,21)(16,25)(19,22)(23,24)Polycyclic presentation:
2^9|b^2=f,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=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, 6399022);