Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628847 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
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 | |
17242 conjugacy classes of subgroups; only the 954 normal subgroups are shown.
This group has 17242 conjugacy classes of subgroups — too many to draw. Only its 954 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)(10,15)(12,13)(16,22)(17,21)(19,20)(23,24);(9,10)(11,16)(12,15)(13,21)(14,17)(18,23)(19,22)(20,24);(2,4)(9,11)(10,16)(12,19)(13,20)(14,18)(15,22)(17,23)(21,24);(6,8)(11,18)(16,23)(19,20)(22,24);(9,12,14,13)(10,15,17,21)(11,19,18,20)(16,22,23,24)Polycyclic presentation:
2^9|e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fi,f^b=fi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72249565, #72249710, #72256883, … |
| maximal normal subgroup of | #72249565, #72249710, #72256883, … |
| maximal quotient of | #72231572, #72264497, #72268357, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536045);