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×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#547 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1183 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,12)(8,19,22,32)(9,14)(10,24,26,15)(11,16)(13,21)(17,33,20,36)(18,31,23,28)(25,30)(27,34)(29,35);(1,2)(3,4)(5,7,11,9)(6,12,16,14)(8,20,22,17)(10,18,26,23)(13,29,30,27)(15,28,24,31)(19,36,32,33)(21,35,25,34);(5,8,11,22)(6,13,16,30)(7,17,9,20)(10,25,26,21)(12,27,14,29)(15,32,24,19)(18,35,23,34)(28,36,31,33);(2,4)(5,9,11,7)(6,14,16,12)(8,17,22,20)(10,23,26,18)(13,27,30,29)(15,31,24,28)(19,33,32,36)(21,34,25,35)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,d^c=di,f^c=fi,g^c=ghi,g^f=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72312406, #72312412, #72312419, … |
| maximal normal subgroup of | #72312406, #72312412, #72312419, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404506);