Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13512 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2085 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,18,22,36)(8,24,27,32)(9,19)(10,15)(11,16)(12,20,31,13)(14,28)(17,30)(21,35,23,26)(25,29)(33,34);(5,7,9,20,10,23,28,24,11,22,29,13,30,21,34,32)(6,12,14,18,17,27,19,35,16,31,33,36,15,8,25,26);(1,2,3,4)(5,8,11,27)(6,13)(7,19)(9,26,29,35)(10,12,30,31)(14,21)(15,24)(16,20)(17,32)(18,34,36,28)(22,25)(23,33)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=eh,d^2=h,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239848, #72239850, #72240850, … |
| maximal normal subgroup of | #72239848, #72239850, #72240850, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56418);