Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30236 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2644 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
952 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,14,10,12,23,15)(7,16,19,11,20,26,17,25)(8,13,21,29,22,27,34,28)(18,30,33,24,32,36,31,35);(1,2)(3,4)(5,7)(6,11,12,25)(8,18)(9,17)(10,20)(13,24,27,35)(14,16,15,26)(19,23)(21,31)(22,32)(28,36,29,30)(33,34);(2,4)(6,12)(7,17)(8,21)(9,23)(11,16)(13,28)(18,32)(19,20)(22,34)(24,35)(25,26)(27,29);(5,8,9,21,10,22,23,34)(6,13,14,29,12,27,15,28)(7,18,17,31,20,32,19,33)(11,24,16,30,25,35,26,36)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287385);