Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23444 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×#534 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
5854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,17,19,14)(12,15)(13,16)(18,21,24,22)(20,23);(2,4)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,24)(21,23);(6,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,23)(17,22)(19,24);(12,20)(15,23)(18,24)(21,22)Polycyclic presentation:
2^9|a^2=h,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^b=hi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72235904, #72236375, #72238085, … |
| maximal normal subgroup of | #72235904, #72236375, #72238085, … |
| maximal quotient of | #72235847, #72236453, #72238049, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264726);