Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480884 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#546 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1260 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)(7,11)(8,18,21,30)(9,22,24,13)(10,14)(12,20)(15,31,19,36)(16,33,35,26)(17,27)(23,29)(25,32)(28,34);(1,2)(3,4)(5,7,10,17)(6,11,14,27)(8,19,21,15)(9,16,24,35)(12,28,29,25)(13,26,22,33)(18,36,30,31)(20,34,23,32);(5,8,10,21)(6,12,14,29)(7,15,17,19)(9,23,24,20)(11,25,27,28)(13,30,22,18)(16,34,35,32)(26,36,33,31);(2,4)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,f^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,f^c=fi,g^c=ghi,g^f=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72317544, #72317572, #72317576, … |
| maximal normal subgroup of | #72317544, #72317572, #72317576, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404563);