Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(10,16)(11,19)(13,14)(18,23)(20,24)(21,22);(2,4)(5,7,8,6)(9,10,15,18)(11,17,20,12)(13,23,14,16)(19,21,24,22);(9,11)(10,17)(12,18)(13,19)(14,24)(15,20)(16,22)(21,23);(9,12,15,17)(10,11,18,20)(13,22,14,21)(16,24,23,19)Polycyclic presentation:
2^9|a^2=h,b^2=hi,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,e^b=ei,g^b=gi,f^c=fi,g^c=gi,e^d=ei,f^d=fi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72271518, #72271523, #72271527 |
| maximal normal subgroup of | #72271518, #72271523, #72271527 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299797);