Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13442 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2054 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#509 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,15)(7,18)(8,12,9,25)(11,28,24,17)(13,21)(14,26,16,22)(19,33)(20,31)(23,32)(27,30)(29,35)(34,36);(2,4)(6,12)(7,19)(8,22)(9,26)(11,24)(13,31)(14,17)(15,25)(16,28)(18,35)(20,23)(21,36)(27,29)(30,32);(5,7)(6,13)(8,23)(9,27)(10,21)(11,29)(12,30)(14,33)(15,18)(16,34)(17,35)(19,26)(20,24)(22,36)(25,32)(28,31)Polycyclic presentation:
2^9|a^2=g,d^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,h^a=hi,c^b=cf,d^b=dh,f^b=fh,h^b=hi,d^c=dhi,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250226, #72250230, #72250231, … |
| maximal normal subgroup of | #72250226, #72250230, #72250231, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55128);