Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394335 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2440 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)(12,17)(13,16)(14,18)(15,21)(20,23)(22,24);(9,11,12,21,14,20,13,22)(10,15,16,19,18,24,17,23);(2,4)(10,16)(11,20)(12,13)(15,19)(17,18)(23,24);(6,8)Polycyclic presentation:
2^9|a^2=ef,b^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72322967 |
| maximal normal subgroup of | #72322967 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302892);