Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213760 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1169 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(9,10)(11,16,19,14)(12,20)(13,17)(15,21)(18,23,24,22);(1,2,4,5)(3,8,7,6)(9,11)(10,14)(12,18)(13,19)(15,23)(16,17)(20,22)(21,24);(9,12)(10,15,17,20)(11,18)(13,21)(14,22,16,23)(19,24);(1,3,4,7)(2,6,5,8)(9,11)(10,16,17,14)(12,18)(13,19)(15,23,20,22)(21,24)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72268076 |
| maximal normal subgroup of | #72268076 |
| maximal quotient of | #72268081, #72268110 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294302);