Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| 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 | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | #1999 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(6,9)(7,11)(10,12)(13,14,17,21,18,22,26,20)(15,23,16,19,24,27,25,28);(5,6,7,9,8,10,11,12)(13,15,17,16,18,24,26,25)(14,19,20,23,22,28,21,27);(2,4)(6,10)(9,12);(6,10)(9,12)(13,16,17,24,18,25,26,15)(14,19,21,27,22,28,20,23)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=g,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=chi,d^b=dh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72224967, #72285170, #72286341, … |
| maximal normal subgroup of | #72224967, #72285170, #72286341, … |
| maximal quotient of | #72224964, #72224965, #72224966, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 118164);