Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129430 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C22 | #1799 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
802 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,14,11,16,8,22)(7,17,20,28,21,33,18,12)(9,15,23,29,24,13,25,30)(19,32,34,27,35,31,36,26);(1,2)(3,4)(5,7,8,18,11,21,10,20)(6,12,14,17,16,28,22,33)(9,19,23,34,24,35,25,36)(13,26,29,31,15,27,30,32);(2,4)(6,13)(9,24)(12,26)(14,29)(15,16)(17,31)(19,35)(22,30)(23,25)(27,28)(32,33)(34,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,c^b=cf,e^c=ei,g^c=ghi,g^e=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55212);