Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
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 ◁ C22×C22⋊C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,14,19,22,17,21,18,16)(15,20,26,23,25,28,27,24);(6,10)(11,12)(13,14,18,16,17,21,19,22)(15,23,26,28,25,24,27,20);(2,4)(5,7,9,8)(6,11,10,12)(13,15,17,25)(14,20,21,28)(16,24,22,23)(18,27,19,26);(5,8,9,7)(6,11,10,12)(13,16,18,21,17,22,19,14)(15,24,27,28,25,23,26,20)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=e,d^2=eh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216189, #72317237, #72317285, … |
| maximal normal subgroup of | #72216189, #72317237, #72317285, … |
| maximal quotient of | #72216074 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 286075);