Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188764 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2283 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,17)(8,22,25,26)(9,12)(10,15)(11,16)(13,21)(14,24,30,23)(18,27)(19,28,32,36)(20,29)(31,34,35,33);(6,12,16,15)(7,18)(13,27,29,17)(14,26,30,22)(19,32)(20,21)(28,35,36,31)(33,34);(1,2,3,4)(5,7,10,21,11,18,9,20)(6,13,12,17,16,29,15,27)(8,19,24,34,25,32,23,33)(14,28,26,31,30,36,22,35);(5,8,9,23,11,25,10,24)(6,14,15,22,16,30,12,26)(7,19,20,33,18,32,21,34)(13,28,27,35,29,36,17,31)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,c^b=ci,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72240728, #72240734, #72264443, … |
| maximal normal subgroup of | #72240728, #72240734, #72264443, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299724);