Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135848 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1853 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
658 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)(5,6,11,16,10,15,12,29)(7,17,20,27,19,28,14,9)(8,21,26,30,24,13,25,31)(18,33,35,23,34,32,22,36);(5,7,9,15,12,20,28,29,10,19,27,6,11,14,17,16)(8,22,23,31,25,18,33,21,24,35,36,30,26,34,32,13);(2,4)(5,8)(6,13)(7,18)(9,23)(10,24)(11,26)(12,25)(14,22)(15,21)(16,31)(17,32)(19,34)(20,35)(27,36)(28,33)(29,30)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,f^b=fi,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250558, #72250568, #72250572, … |
| maximal normal subgroup of | #72250558, #72250568, #72250572, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55487);