Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #400512 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2491 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15004 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
796 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)(3,7,13,21)(4,8)(5,9)(6,10)(11,18)(12,26,15,19)(14,20,29,22)(16,24)(17,23)(25,27)(28,32,30,31);(1,3,5,14,6,13,18,29)(2,7,11,22,10,21,9,20)(4,15,17,30,16,12,27,28)(8,19,25,32,24,26,23,31);(1,4)(2,8)(3,12,13,15)(5,17)(6,16)(7,19,21,26)(9,23)(10,24)(11,25)(14,28,29,30)(18,27)(20,31,22,32);(2,9,10,11)(7,20,21,22)(8,23,24,25)(19,31,26,32)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=f,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319815 |
| maximal normal subgroup of | #72319815 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 316142);