Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,17,12,18,10,16)(7,19,21,13,22,14,8,23)(9,15,25,31,26,33,27,32)(20,30,24,36,34,29,35,28);(1,2)(3,4)(5,7,12,22)(6,13,18,23)(8,10,21,11)(9,24,26,35)(14,17,19,16)(15,30,33,29)(20,27,34,25)(28,32,36,31);(2,4)(5,8,10,22,12,21,11,7)(6,14,17,23,18,19,16,13)(9,24,25,34,26,35,27,20)(15,29,32,36,33,30,31,28);(5,9,10,25,12,26,11,27)(6,15,16,31,18,33,17,32)(7,20,8,24,22,34,21,35)(13,28,19,30,23,36,14,29)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291529);