Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393904 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(5,6)(7,8)(9,10)(11,19)(12,18)(13,16)(14,17)(15,20)(21,24)(22,23);(2,4)(9,11)(10,15)(12,22)(13,20)(14,21)(16,24)(17,23)(18,19);(10,16)(11,20)(12,13)(15,23)(17,18)(21,22);(6,8)(10,17)(11,21)(16,18)(20,22)Polycyclic presentation:
2^9|a^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72256520, #72256521, #72256523 |
| maximal normal subgroup of | #72256520, #72256521, #72256523 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302579);