Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13398 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #481 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
796 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)(10,15,16,23)(11,12)(13,22)(17,24,21,18)(19,20);(1,2,3,4)(9,10,11,15,13,18,12,21,14,16,20,23,22,24,19,17);(6,8)(10,15)(11,19)(12,20)(13,22)(16,23)(17,18)(21,24)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,d^c=dhi,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216645, C2×#148388, #72224843, … |
| maximal normal subgroup of | #72216645, C2×#148388, #72224843, … |
| maximal quotient of | #72216573, #72216581, C2×#148388 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55584);