Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #205795 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#390 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1271 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)(3,5,7,8)(4,6)(9,10,13,17,14,18,12,15)(11,19,22,24,20,23,21,16);(1,3)(2,5,6,8)(4,7)(10,15,18,17)(16,23,24,19);(9,11,12,21,14,20,13,22)(10,16,17,19,18,24,15,23);(11,20)(16,24)(19,23)(21,22)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=f,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,e^b=ei,g^b=gi,d^c=dhi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 113263);