Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457252 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16291 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,13,6,24)(3,14,18,28,19,27,17,9)(4,20,5,11,23,30,22,10)(15,25,21,26,16,31,32,29);(1,3)(2,9,13,28)(4,21)(5,16)(6,17)(7,18)(8,19)(10,29,11,26)(12,14,24,27)(15,22)(20,25,30,31)(23,32);(1,4)(2,10)(3,15)(5,6)(7,22)(8,23)(9,25)(11,13)(12,30)(14,29)(16,19)(17,21)(18,32)(20,24)(26,27)(28,31);(1,5,7,23,8,22,6,4)(2,11,24,20,13,10,12,30)(3,16,18,32,19,15,17,21)(9,26,27,25,28,29,14,31)Polycyclic presentation:
2^9|a^2=f,b^2=ef,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237071, #72237072, #72237082, … |
| maximal normal subgroup of | #72237071, #72237072, #72237082, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364449);