Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455697 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16268 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23400 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1124 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)(3,11)(4,9)(5,18)(6,8)(7,15)(10,12)(13,26)(14,29)(16,23)(17,22)(19,24)(20,27)(21,31)(25,28)(30,32);(1,3,12,27,6,13,5,15)(2,7,18,26,8,20,10,11)(4,14,16,31,17,28,19,30)(9,21,24,29,22,32,23,25);(2,8)(3,13)(4,16)(5,12)(7,20)(9,23)(14,30)(17,19)(21,25)(22,24)(28,31)(29,32);(1,4)(2,9)(3,14)(5,19)(6,17)(7,21)(8,22)(10,23)(11,25)(12,16)(13,28)(15,30)(18,24)(20,32)(26,29)(27,31)Polycyclic presentation:
2^9|b^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,e^d=ei,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237069, #72237078, #72237092, … |
| maximal normal subgroup of | #72237069, #72237078, #72237092, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362893);