Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #403928 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2615 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14885 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2576 conjugacy classes of subgroups; only the 240 normal subgroups are shown.
This group has 2576 conjugacy classes of subgroups — too many to draw. Only its 240 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13)(4,18)(5,12)(6,10)(7,11)(8,15)(9,19)(14,25)(16,24)(17,23)(20,26)(21,27)(22,31)(28,30)(29,32);(1,3)(2,8)(4,14)(5,16)(6,15)(7,17)(9,25)(10,23)(11,24)(12,13)(18,32)(19,30)(20,31)(21,29)(22,27)(26,28);(1,4)(2,9)(3,14)(5,21)(6,19)(7,20)(8,22)(10,26)(11,27)(12,18)(13,28)(15,30)(16,29)(17,31)(23,32)(24,25);(2,10,11,12)(8,23,24,13)(9,26,27,18)(22,32,25,28)Polycyclic presentation:
2^9|d^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285237, #72285398, #72286033, … |
| maximal normal subgroup of | #72285237, #72285398, #72286033, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311124);