Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #205799 |
| 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 | |
1329 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)(11,19)(12,15)(13,23)(14,18)(16,21)(17,20)(22,24);(1,3,4,7)(2,5)(6,8)(9,11,20,22,14,21,13,12)(10,15,23,16,18,24,17,19);(9,12,13,21,14,22,20,11)(10,16,17,15,18,19,23,24);(3,7)(5,8)(11,21)(12,22)(15,24)(16,19)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,e^2=hi,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,g^b=gi,d^c=dhi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72339517, #72339519, #72339521, … |
| maximal normal subgroup of | #72339517, #72339519, #72339521, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112996);