Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5020720 |
| automorphism | Aut(G) order 25165824 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5850 conjugacy classes of subgroups; only the 658 normal subgroups are shown.
This group has 5850 conjugacy classes of subgroups — too many to draw. Only its 658 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(9,10,12,14)(11,15,16,13)(17,18)(19,21)(20,22)(23,24);(9,10)(11,13)(12,14)(15,16)(17,18,21,19)(20,22,24,23);(2,3)(11,16)(13,15)(18,19)(20,24);(1,3,4,2)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,18)(20,23,24,22);(9,11,12,16)(10,13,14,15)(17,20,21,24)(18,22,19,23)Polycyclic presentation:
2^9|a^2=gi,b^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213929, #72213963, #72219740, … |
| maximal normal subgroup of | #72213929, #72213963, #72219740, … |
| maximal quotient of | #72222724, #72222731, C2×#5021378 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4928574);