Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29730 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,21,19,22,17,15)(16,20,24,27,25,28,26,23);(6,9)(10,12)(13,15,17,22,19,21,18,14)(16,20,24,27,25,28,26,23);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,16)(14,20)(15,23)(17,26)(18,24)(19,25)(21,27)(22,28);(7,11)(10,12)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,d^c=de,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72317267, #72317286, #72317288, … |
| maximal normal subgroup of | #72317267, #72317286, #72317288, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287977);