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 | |
446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6)(7,17,21,27)(8,12)(9,14)(10,24)(11,16)(13,20,31,19)(15,26)(18,29,36,33)(22,32)(23,28)(25,30,35,34);(1,2,4,3)(6,12)(7,18,21,36)(8,9)(10,22)(13,29,31,33)(14,16)(15,24)(17,34,27,30)(19,25,20,35)(23,26);(1,3,4,2)(5,7,8,19,11,21,9,20)(6,13,14,17,16,31,12,27)(10,25,22,18,26,35,23,36)(15,33,32,34,24,29,28,30)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=d,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,h^b=hi,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72240847, #72240848, #72240850, … |
| maximal normal subgroup of | #72240847, #72240848, #72240850, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56159);