Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32929 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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 | C22×C4⋊C4 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,23,18,21,17,27)(15,24,26,28,25,22,16,20);(5,7,8,11)(6,9,10,12)(13,15,19,26,18,25,17,16)(14,20,27,22,21,28,23,24);(6,10)(9,12)(14,21)(20,28)(22,24)(23,27);(2,4)(13,16,17,25,18,26,19,15)(14,22,23,20,21,24,27,28)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi152 conjugacy classes — view the full character table.
| maximal subgroup of | #72317189, #72317199, #72317201, … |
| maximal normal subgroup of | #72317189, #72317199, #72317201, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299615);