Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13323 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1999 |
| 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 | |
1010 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)(5,6,10,17,11,15,27,18)(7,19,22,34,23,13,28,33)(8,16,9,14,26,12,24,30)(20,31,21,32,25,29,36,35);(6,12)(7,20)(8,24)(9,26)(10,27)(13,19)(14,17)(15,16)(18,30)(21,28)(22,36)(23,25)(29,32)(31,35);(2,4)(5,7)(6,13)(8,25)(9,21)(10,28)(11,23)(12,29)(14,35)(15,19)(16,31)(17,34)(18,33)(20,26)(22,27)(24,36)(30,32)Polycyclic presentation:
2^9|a^2=g,d^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,h^a=hi,c^b=cf,d^b=dhi,f^b=fh,g^b=gi,h^b=hi,d^c=dh,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250029, #72250041, #72250050, … |
| maximal normal subgroup of | #72250029, #72250041, #72250050, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55126);