Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2749 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14376 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
981 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)(3,7,13,23)(4,14,16,8)(5,17)(6,9)(10,20)(11,25,18,21)(12,28)(15,24,19,26)(22,30)(27,31)(29,32);(3,11,13,18)(7,21,23,25)(12,29,30,27)(22,32,28,31);(1,3)(2,7,9,23)(4,12)(5,18)(6,13)(8,22,14,28)(10,25,17,21)(11,20)(15,27)(16,30)(19,29)(24,31,26,32);(1,4)(2,8)(3,12)(5,19)(6,16)(7,22)(9,14)(10,26)(11,27)(13,30)(15,20)(17,24)(18,29)(21,31)(23,28)(25,32)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=ef,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285906, #72286090, #72286218 |
| maximal normal subgroup of | #72285906, #72286090, #72286218 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291488);