Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6629016 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×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 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(10,14)(12,15)(16,20)(17,21)(19,22)(23,24);(9,10,15,21,13,17,12,14)(11,16,22,24,18,23,19,20);(2,4)(9,11)(10,16)(12,19)(13,18)(14,20)(15,22)(17,23)(21,24);(6,8)(10,17)(14,21)(16,23)(20,24);(11,18)(16,23)(19,22)(20,24)Polycyclic presentation:
2^9|b^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,d^b=di,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72249635, #72249643, #72256889 |
| maximal normal subgroup of | #72249635, #72249643, #72256889 |
| maximal quotient of | #72264783, #72310617, #72311588 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536213);