Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499724 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
882 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)(5,6)(7,11)(8,9)(10,12);(6,9)(10,11)(13,14)(15,20)(16,19)(17,26)(18,22)(21,24)(23,27)(25,28);(2,4)(14,19)(15,17)(16,24)(20,21)(22,28)(23,25)(26,27);(5,7,8,12)(6,10,9,11)(13,15,18,17)(14,20,22,26)(16,23,25,24)(19,27,28,21)Polycyclic presentation:
2^9|d^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,d^c=di,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319285, #72319515, #72319560, … |
| maximal normal subgroup of | #72319285, #72319515, #72319560, … |
| maximal quotient of | #72322616, #72322816 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407645);