Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417454 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35659 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3350 conjugacy classes of subgroups; only the 262 normal subgroups are shown.
This group has 3350 conjugacy classes of subgroups — too many to draw. Only its 262 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 — 4 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(9,10,15,20)(11,17,12,22)(13,18,19,14)(16,24,23,21);(9,11,15,12)(10,16,20,23)(13,17,19,22)(14,24,18,21);(2,4)(9,12,15,11)(10,17,20,22)(13,23,19,16)(14,21,18,24);(6,8)(9,13)(10,18)(11,16)(12,23)(14,20)(15,19)(17,24)(21,22)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72258239, #72258259, #72258264 |
| maximal normal subgroup of | #72258239, #72258259, #72258264 |
| maximal quotient of | #72267308 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356682);