Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470237 |
| 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 | |
4520 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)(3,5)(4,6)(7,8)(12,13)(15,18)(16,22)(19,20);(1,3)(2,5)(4,7)(6,8)(9,10)(11,18)(12,22)(13,16)(14,17)(15,21)(19,23)(20,24);(3,7)(5,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24);(3,7)(5,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24);(2,6)(5,8)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)Polycyclic presentation:
2^9|d^2=i,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dh,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72240649, #72252418, #72252430, … |
| maximal normal subgroup of | #72240649, #72252418, #72252430, … |
| maximal quotient of | #72251707, #72267161 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377437);