Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#120 |
| prime radical | #23406 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,15,12,16,10,28)(7,17,20,13,22,30,21,31)(8,23,25,14,26,9,24,27)(18,34,35,32,36,29,33,19);(1,2)(3,4)(5,7)(6,13)(8,18)(9,19)(10,21)(11,20)(12,22)(14,29)(15,30)(16,31)(17,28)(23,32)(24,33)(25,35)(26,36)(27,34);(2,4)(5,8)(6,14)(7,18)(9,15)(10,24)(11,25)(12,26)(13,29)(16,27)(17,32)(19,30)(20,35)(21,33)(22,36)(23,28)(31,34);(5,9,11,27,12,23,10,14)(6,8,15,25,16,26,28,24)(7,19,20,34,22,32,21,29)(13,18,30,35,31,36,17,33)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^c=fi,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72321148, #72322108 |
| maximal normal subgroup of | #72321148, #72322108 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283020);