Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13334 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| 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 | |
1206 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)(7,18)(8,12)(9,25)(10,28)(11,16)(13,19)(14,22)(15,26)(17,24)(20,33)(21,30)(23,32)(27,35)(29,34)(31,36);(6,12)(8,22)(9,26)(10,24)(13,18)(14,17)(15,28)(16,25)(19,27)(20,29)(23,36)(30,31)(32,33)(34,35);(5,7)(6,13)(8,23)(9,27)(10,29)(11,21)(12,30)(14,33)(15,34)(16,31)(17,35)(18,25)(19,26)(20,24)(22,36)(28,32)Polycyclic presentation:
2^9|a^2=g,d^2=hi,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=dh,e^b=eh,h^b=hi,d^c=dhi,e^c=eh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231499, #72231546, #72231547, … |
| maximal normal subgroup of | #72231499, #72231546, #72231547, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55080);