Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20671 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2283 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1482 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)(7,16,19,12)(8,13)(9,11)(10,15)(14,22)(17,29,30,25)(18,26,32,23)(20,28)(21,24)(27,33)(31,35,36,34);(6,11,15,14)(8,20)(12,23,16,26)(13,27,28,24)(17,30)(21,33)(25,35,29,34)(31,36);(2,4)(5,7)(6,12)(8,17)(9,18)(10,19)(11,23)(13,25)(14,26)(15,16)(20,30)(21,31)(22,32)(24,34)(27,35)(28,29)(33,36);(5,8,9,21,10,20,22,33)(6,13,11,24,15,28,14,27)(7,17,18,31,19,30,32,36)(12,25,23,34,16,29,26,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=eh,e^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,d^b=di,g^c=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72231520, #72231526, #72231553, … |
| maximal normal subgroup of | #72231520, #72231526, #72231553, … |
| maximal quotient of | #72265314, #72280575, C2×#354934, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262130);