Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #248647 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2069 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)(3,5)(4,6)(7,8)(9,10)(11,19)(12,15)(13,23)(14,18)(16,20)(17,21)(22,24);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,21)(13,22)(14,20)(16,17)(18,24)(19,23);(5,8)(11,20)(13,21)(15,24)(17,23);(9,12,14,22)(10,16,18,19)(11,21,20,13)(15,17,24,23)Polycyclic presentation:
2^9|b^2=hi,d^2=h,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fi,e^c=eh,e^d=eh,f^d=fh53 conjugacy classes — view the full character table.
| maximal subgroup of | #72340999, #72341001 |
| maximal normal subgroup of | #72341001 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 155863);