Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
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)(19,25)(23,28)(24,27);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(15,23)(19,27)(22,28)(24,25);(2,4)(13,15,17,25,18,23,16,24)(14,19,20,22,21,27,26,28);(2,4);(13,16,18,17)(14,20,21,26)(15,24,23,25)(19,22,27,28)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=dhi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72249945, #72249960, #72256834, … |
| maximal normal subgroup of | #72249945, #72249960, #72256834, … |
| maximal quotient of | #72265594, #72307867 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350632);