Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394273 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1461 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,13)(12,15)(14,16)(17,24)(18,22)(19,20)(21,23);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,19,21,17)(14,18)(15,20,23,24)(16,22);(1,4,5,3)(2,6,7,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,23)(18,19)(22,24);(9,13)(10,11)(12,20)(14,22)(15,17)(16,18)(19,23)(21,24)Polycyclic presentation:
2^9|a^2=e,b^2=eh,c^2=e,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319655, #72320090, #72320194 |
| maximal normal subgroup of | #72319655, #72320090, #72320194 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312732);