Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| 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×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1843 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(13,14)(15,20)(16,22)(17,23)(18,27)(19,24)(21,25)(26,28);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15,19,16)(14,20,24,22)(17,26,18,25)(21,23,28,27);(1,3,4,2)(6,9)(7,11)(13,15)(14,21)(16,19)(17,25)(18,26)(20,27)(22,23)(24,28);(5,7,8,11)(6,10,9,12)(13,16,19,15)(14,22,24,20)(17,25,18,26)(21,27,28,23);(6,9)(10,12)(13,17,19,18)(14,23,24,27)(15,25,16,26)(20,21,22,28)Polycyclic presentation:
2^9|b^2=fi,c^2=f,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,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, 6421633);