Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#397 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,16,11,14,9,15)(7,17,19,29,21,12,20,28)(8,22,25,31,24,32,26,13)(18,34,36,33,35,30,23,27);(1,2)(3,4)(5,7)(6,12)(8,23)(9,19)(10,20)(11,21)(13,30)(14,17)(15,28)(16,29)(18,26)(22,33)(24,36)(25,35)(27,32)(31,34);(2,4)(5,8)(6,13,14,31)(7,18)(9,25)(10,26)(11,24)(12,27,17,33)(15,22,16,32)(19,23)(20,36)(21,35)(28,30,29,34);(8,24)(13,31)(18,35)(22,32)(23,36)(25,26)(27,33)(30,34)Polycyclic presentation:
2^9|a^2=h,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,g^b=gi,h^b=hi,d^c=di,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250584, #72250600, #72250614, … |
| maximal normal subgroup of | #72250584, #72250600, #72250614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265126);