Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,28)(25,27)(29,33)(30,34)(31,36)(32,35);(6,9)(10,12)(21,23,29,30,25,31,24,32)(22,26,33,34,27,36,28,35);(5,7)(6,10)(8,11)(9,12)(14,17)(18,20)(23,30)(24,29)(26,34)(28,33)(31,32)(35,36);(7,11)(10,12)(13,15,16,19)(14,18,17,20)(22,27)(23,31)(28,33)(30,32)Polycyclic presentation:
2^9|a^2=h,b^2=gi,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=di,d^c=de,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72240762, #72240768, #72240778, … |
| maximal normal subgroup of | #72240762, #72240768, #72240778, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288089);