Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394020 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,18)(12,22)(13,16)(14,17)(15,19)(20,23)(21,24);(1,3,5,4)(2,6,7,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,23)(21,24);(1,4,5,3)(2,6,7,8)(9,11)(10,15)(12,20)(13,19)(14,21)(16,24)(17,23)(18,22);(9,10)(11,15)(12,22)(13,16)(14,17)(18,20)(19,24)(21,23)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72261241, #72261262, #72282800, … |
| maximal normal subgroup of | #72261241, #72261262, #72282800, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302068);