Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394463 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#393 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,18,20,24)(12,16)(13,22)(14,17)(15,21,23,19);(2,5)(6,7)(11,19,20,21)(15,18,23,24);(9,11)(10,15)(12,19)(13,21)(14,20)(16,24)(17,23)(18,22);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^9|a^2=e,b^2=eh,d^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^c=ei,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72224748, #72224757 |
| maximal normal subgroup of | #72224748, #72224757 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 323211);