Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1792 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18,21,24)(19,20,23,22);(5,6,8,7)(10,13)(14,16)(17,19,21,23)(18,22,24,20);(2,4)(18,23)(19,24)(20,22);(5,7,8,6)(9,11,12,15)(10,14,13,16)(17,20,21,22)(18,23,24,19)Polycyclic presentation:
2^9|a^2=i,b^2=hi,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72265547, #72265563, #72265624, … |
| maximal normal subgroup of | #72265547, #72265563, #72265624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312798);