Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14366 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1517 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,8,3,7,11)(4,15,17,9,18,25,23,13)(5,10,21,26,22,14,20,19)(16,29,32,27,31,28,30,24);(1,3)(2,8)(4,13,18,9)(5,19)(6,11)(7,12)(10,21)(14,20)(15,23,25,17)(16,28,31,29)(22,26)(24,30,27,32);(1,4)(2,9)(3,13)(5,16)(6,17)(7,23)(8,18)(10,24)(11,15)(12,25)(14,27)(19,28)(20,30)(21,32)(22,31)(26,29);(1,5)(2,10)(3,14)(4,16)(6,21)(7,20)(8,22)(9,24)(11,19)(12,26)(13,27)(15,28)(17,32)(18,31)(23,30)(25,29)Polycyclic presentation:
2^9|a^2=h,b^2=ef,e^2=i,f^2=i,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,e^b=ei,f^b=fi,g^b=gi,f^c=fi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72318583, #72318864, #72318894, … |
| maximal normal subgroup of | #72318583, #72318864, #72318894, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291123);