Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232169 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1236 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)(7,17,20,11)(8,21)(9,16)(10,14)(12,24)(13,19,29,26)(15,27)(18,31,35,30)(22,28,36,34)(23,33)(25,32);(1,2)(3,4)(5,7)(6,11)(8,22)(9,26)(10,20)(12,30)(13,15)(14,17)(16,29)(18,25)(19,27)(21,28)(23,36)(24,35)(31,32)(33,34);(2,4)(5,8,9,24,10,23,27,25)(6,12,15,21,14,32,16,33)(7,18,26,22,20,35,19,36)(11,28,13,31,17,34,29,30);(5,7)(6,13)(8,18)(9,26)(10,20)(11,16)(12,31)(14,29)(15,17)(19,27)(21,34)(22,24)(23,35)(25,36)(28,33)(30,32)Polycyclic presentation:
2^9|a^2=h,c^2=fh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296189);