Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219508 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1411 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,15)(7,16)(8,13,22,31)(9,11,23,14)(12,19)(17,32)(18,25)(20,29)(21,26,35,30)(24,27)(28,33)(34,36);(6,11)(9,23)(12,27)(13,26)(14,15)(16,25)(18,24)(20,34)(21,35)(28,36)(29,32)(30,31);(2,4)(5,7)(6,12)(8,20)(9,24)(10,19)(11,25)(13,29)(14,27)(15,16)(17,21)(18,23)(22,34)(26,32)(28,30)(31,36)(33,35);(5,8)(6,13)(7,17,19,33)(9,21)(10,22)(11,26)(12,28,16,32)(14,30)(15,31)(18,20,24,34)(23,35)(25,29,27,36)Polycyclic presentation:
2^9|a^2=h,d^2=hi,e^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,d^c=de,e^c=ei,h^c=hi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72256504, #72256515, #72256521 |
| maximal normal subgroup of | #72256504, #72256515, #72256521 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288671);