Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417508 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | #35347 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3155 conjugacy classes of subgroups; only the 262 normal subgroups are shown.
This group has 3155 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)(11,14)(12,15)(13,16)(17,21)(18,22)(19,24)(20,23);(9,11,13,18)(10,14,16,22)(12,19)(15,21)(17,20)(23,24);(2,4)(9,12)(10,15)(11,17)(13,20)(14,21)(16,23)(18,19)(22,24);(6,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,24)(20,23)Polycyclic presentation:
2^9|b^2=ei,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72258151, #72311812 |
| maximal normal subgroup of | #72258151, #72311812 |
| maximal quotient of | #72258158, #72258163, #72312805 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325018);