Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10)(11,19)(12,18)(13,16)(14,17)(15,21)(20,23)(22,24);(2,4)(5,7,8,6)(9,11)(10,15)(12,22)(13,21)(14,20)(16,23)(17,24)(18,19);(10,16)(12,13)(15,23)(17,18)(19,24)(21,22);(11,20)(15,24)(19,23)(21,22);(10,17)(15,24)(16,18)(19,23)Polycyclic presentation:
2^9|a^2=h,b^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72307408, #72307502, #72307635, … |
| maximal normal subgroup of | #72307408, #72307502, #72307635, … |
| maximal quotient of | #72310240, #72315994, #72316115, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278518);