Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384252 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14375 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1074 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)(10,18)(11,25,19,21)(12,28)(15,24,20,26)(22,30)(27,31)(29,32);(3,11)(5,18)(7,21)(10,17)(12,29)(13,19)(15,20)(22,32)(23,25)(24,26)(27,30)(28,31);(1,3)(2,7,9,23)(4,12)(5,19)(6,13)(8,22,14,28)(10,25,17,21)(11,18)(15,27)(16,30)(20,29)(24,31,26,32);(1,4)(2,8)(3,12)(5,20)(6,16)(7,22)(9,14)(10,26)(11,27)(13,30)(15,18)(17,24)(19,29)(21,31)(23,28)(25,32)Polycyclic presentation:
2^9|a^2=h,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,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,g^d=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291448);