Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2858 conjugacy classes of subgroups.
| 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)(11,15)(12,16)(13,21)(14,17)(18,22)(19,23)(20,24);(9,10)(11,18,20,23)(12,21)(13,16)(14,17)(15,19,24,22);(2,4)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=di,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72265528, #72265536, #72265574, … |
| maximal normal subgroup of | #72265528, #72265536, #72265574, … |
| maximal quotient of | #72279808 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264940);