Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140598 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1961 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C2×#3155 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1202 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)(7,17,20,36)(8,22)(9,13)(10,28)(11,15)(12,18,32,26)(14,25)(16,27)(19,33,23,35)(21,31,29,30)(24,34);(2,4)(5,7)(6,12)(8,23)(9,26)(10,29)(11,20)(13,31)(14,17)(15,32)(16,35)(18,24)(19,25)(21,27)(22,36)(28,33)(30,34);(6,13)(8,24)(9,25)(10,27)(12,30)(14,16)(15,34)(17,33)(18,23)(19,26)(21,29)(22,28)(31,32)(35,36)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,g^b=gi,h^b=hi,d^c=dhi,e^c=eh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231505, #72231517, #72231532, … |
| maximal normal subgroup of | #72231505, #72231517, #72231532, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55083);