Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245567 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
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 | |
1259 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(9,10)(11,18)(12,14)(13,17)(15,20)(16,21)(19,23)(22,24);(1,2,5,7)(3,6,4,8)(10,14)(11,19)(12,21)(15,23)(16,17)(18,24);(1,3,5,4)(2,6,7,8)(12,21)(14,16)(18,23)(20,22);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,22)(13,19)(14,18)(16,23)(17,24)(20,21)Polycyclic presentation:
2^9|b^2=hi,c^2=hi,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,d^b=dh,e^b=eh,e^c=ei,e^d=eh,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72282472 |
| maximal normal subgroup of | #72282472 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152803);