Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,15,11,14,9,23)(7,12,19,27,17,26,18,33)(8,20,21,13,22,29,24,28)(16,30,32,25,31,36,34,35);(1,2,3,4)(5,7)(6,12)(8,16)(9,18)(10,19)(11,17)(13,25)(14,26)(15,27)(20,30)(21,32)(22,31)(23,33)(24,34)(28,35)(29,36);(5,8)(6,13)(7,16)(9,24)(10,21)(11,22)(12,25)(14,28)(15,29)(17,31)(18,34)(19,32)(20,23)(26,35)(27,36)(30,33);(7,17)(12,26)(16,31)(18,19)(25,35)(27,33)(30,36)(32,34);(6,14)(12,26)(13,28)(15,23)(20,29)(25,35)(27,33)(30,36)Polycyclic presentation:
2^9|a^2=h,b^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,d^b=di,g^c=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72341714, #72341724, #72341760, … |
| maximal normal subgroup of | #72341714, #72341724, #72341760, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277999);