Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #481 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1874 |
| prime radical | C25 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1038 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)(7,15,17,32)(8,19)(9,23)(10,13)(11,16,26,33)(12,22)(14,25)(18,34,24,28)(20,27,35,30)(21,31)(29,36);(5,7)(6,11,13,26)(8,20)(9,24)(10,17)(12,28,29,34)(14,30,23,27)(15,31,32,19)(16,22)(18,25)(21,35)(33,36);(2,4)(8,21)(12,29)(15,32)(16,33)(19,31)(20,35)(22,36)(28,34)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fi,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72214470, #72214477, #72216369, … |
| maximal normal subgroup of | #72214470, #72214477, #72216369, … |
| maximal quotient of | #72214240 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53723);