Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4750 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)(7,11)(8,18,20,12)(9,13)(10,14)(15,27,28,22)(16,23)(17,24)(19,31,32,25)(21,26)(29,35,36,33)(30,34);(1,2,3,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,22)(13,24)(14,23)(18,27)(19,29)(20,28)(21,30)(25,33)(26,34)(31,35)(32,36);(5,8)(6,12)(7,15)(9,19)(10,20)(11,22)(13,25)(14,18)(16,28)(17,29)(21,32)(23,27)(24,33)(26,31)(30,36)(34,35);(7,16)(11,23)(15,28)(17,30)(22,27)(24,34)(29,36)(33,35);(5,9,10,21)(6,13,14,26)(7,17,16,30)(8,19,20,32)(11,24,23,34)(12,25,18,31)(15,29,28,36)(22,33,27,35)Polycyclic presentation:
2^9|a^2=h,b^2=f,e^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=di,h^c=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72280769, #72280810, #72280840, … |
| maximal normal subgroup of | #72280769, #72280810, #72280840, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277970);