Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394061 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
5361 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)(11,14)(12,15)(13,16)(17,20)(18,21)(19,23)(22,24);(9,11)(10,14)(12,17)(13,18)(15,22)(16,21)(19,24)(20,23);(2,4)(9,12)(10,15)(11,17)(13,19)(14,20)(16,23)(18,24)(21,22);(6,8)(11,18)(14,21)(17,24)(20,22)Polycyclic presentation:
2^9|b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225689, #72234958, #72234959, … |
| maximal normal subgroup of | #72225689, #72234958, #72234959, … |
| maximal quotient of | #72225643, #72233895, #72236756, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301257);