Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32700 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2760 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14375 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,9,21,25)(3,11,14,26,13,7,19,23)(4,15,18,27,17,8,20,24)(12,28,29,32,16,22,30,31);(1,3)(2,7)(4,16)(5,19)(6,13)(8,22)(9,11)(10,26)(12,17)(14,21)(15,28)(18,29)(20,30)(23,25)(24,32)(27,31);(1,4)(2,8)(3,12,13,16)(5,20)(6,17)(7,22,11,28)(9,15)(10,27)(14,30,19,29)(18,21)(23,32,26,31)(24,25);(7,11)(22,28)(23,26)(31,32)Polycyclic presentation:
2^9|a^2=h,c^2=ef,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284899, #72286050, #72286066, … |
| maximal normal subgroup of | #72284899, #72286050, #72286066, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291454);