Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,11,12,9)(13,14)(15,22)(16,20)(17,26)(18,21)(19,24)(23,28)(25,27);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(15,23)(19,27)(22,28)(24,25);(2,4)(13,15,17,24,18,23,16,25)(14,19,20,22,21,27,26,28);(2,4)(13,14)(15,19)(16,26)(17,20)(18,21)(22,24)(23,27)(25,28);(13,16,18,17)(14,20,21,26)(15,24,23,25)(19,28,27,22)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249936, #72249945, #72250027, … |
| maximal normal subgroup of | #72249936, #72249945, #72250027, … |
| maximal quotient of | #72307867, #72308451 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350677);