Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342836 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C2×#3489 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4482 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)(3,5,7,8)(4,6)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,24)(21,23);(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,18)(22,24);(1,3)(2,5,6,8)(4,7)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24);(11,20)(15,18)(19,21)(23,24);(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)Polycyclic presentation:
2^9|a^2=h,c^2=fgh,e^2=f,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,g^c=gi,h^c=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72339546, #72339548, #72339552, … |
| maximal normal subgroup of | #72339546, #72339548, #72339552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6301235);