Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6629016 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(10,14)(12,15)(16,21)(17,22)(18,23)(20,24);(9,10,15,23,13,18,12,14)(11,17,24,21,19,16,20,22);(1,2,3,4)(9,11)(10,16)(12,20)(13,19)(14,21)(15,24)(17,18)(22,23);(6,8)(9,11)(10,17)(12,20)(13,19)(14,22)(15,24)(16,18)(21,23);(11,19)(16,17)(20,24)(21,22)Polycyclic presentation:
2^9|b^2=fi,c^2=g,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ci,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250084, #72250109, #72250352, … |
| maximal normal subgroup of | #72250084, #72250109, #72250352, … |
| maximal quotient of | #72264563 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6537264);