Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393934 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
4864 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)(14,20)(15,22);(5,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,20)(19,22)(23,24);(2,4)(6,8)(9,11,13,18)(10,14,16,20)(12,17,19,23)(15,21,22,24);(9,12)(10,15)(11,17)(13,19)(14,21)(16,22)(18,23)(20,24)Polycyclic presentation:
2^9|c^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^b=fi,h^b=hi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72225690, #72235463, #72238018, … |
| maximal normal subgroup of | #72225690, #72235463, #72238018, … |
| maximal quotient of | #72219233, #72219238, #72225659, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301263);