Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13340 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #493 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2011 |
| prime radical | C2×#3414 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
986 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)(7,16)(8,22)(9,24)(10,29)(11,15)(12,18)(13,26)(14,25)(17,30)(19,35)(20,31)(21,36)(23,34)(27,32)(28,33);(5,7)(6,12)(8,23)(9,27)(10,30)(11,19)(13,17)(14,34)(15,31)(16,22)(18,26)(20,28)(21,25)(24,35)(29,32)(33,36);(5,6)(7,17)(8,24)(9,22)(10,29)(11,15)(12,27)(13,28)(14,25)(16,21)(18,34)(19,36)(20,32)(23,31)(26,33)(30,35)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,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,h^b=hi,d^c=dhi,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249886, #72249904, #72249923, … |
| maximal normal subgroup of | #72249886, #72249904, #72249923, … |
| maximal quotient of | #72265622, #72280352, #72308760 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55134);