Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501764 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#476 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
592 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 592 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,5,7,8)(9,10,14,24,11,20,22,23,16,12,13,19,15,18,21,17);(1,3,4,7)(2,5,6,8)(9,11,16,15)(10,12)(13,21,14,22)(17,23)(18,20)(19,24);(3,7)(5,8)(9,12)(10,16)(11,20)(13,23)(14,17)(15,18)(19,22)(21,24);(9,13,11,21,16,14,15,22)(10,17,18,19,12,23,20,24)Polycyclic presentation:
2^9|a^2=efg,b^2=egi,d^2=e,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^c=fh,g^c=gi,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72224853 |
| maximal normal subgroup of | #72224853 |
| maximal quotient of | #72224847 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 414620);