Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135932 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #442 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(5,6)(7,8)(10,15,17,22)(11,12)(13,23)(16,18,20,24)(19,21);(1,2,3,4)(10,16)(11,19)(12,21)(13,23)(15,18)(17,20)(22,24);(6,8)(9,10)(11,20)(12,22)(13,24)(14,17)(15,19)(16,21)(18,23)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=ehi,f^b=fh,h^b=hi,e^c=ehi,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249886, #72249912, #72249925, … |
| maximal normal subgroup of | #72249886, #72249912, #72249925, … |
| maximal quotient of | #72264826, #72279642, C2×#148104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55300);