Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437621 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3953 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(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);(2,5)(6,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,23)(18,24)(21,22);(9,12)(10,15)(11,17)(13,19)(14,20)(16,23)(18,24)(21,22);(1,3,4,7)(2,6,5,8)(11,18)(14,21)(17,24)(20,22)Polycyclic presentation:
2^9|e^2=g,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72210515, #72210583, #72213205, … |
| maximal normal subgroup of | #72210515, #72210583, #72213205, … |
| maximal quotient of | C2×#6454047, #72258132, #72262130, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361243);