Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135848 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1853 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,10,16,12,17,11,15)(7,18,20,29,21,28,14,9)(8,13,25,30,26,31,24,33)(19,34,22,23,36,35,32,27);(5,7,9,17,11,20,28,15,12,21,29,6,10,14,18,16)(8,22,23,30,24,36,34,31,26,32,27,33,25,19,35,13);(1,2,3,4)(5,8)(6,13)(7,19)(9,27)(10,25)(11,24)(12,26)(14,32)(15,33)(16,30)(17,31)(18,34)(20,22)(21,36)(23,29)(28,35)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=e,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,g^b=gi,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250548, #72250560, #72250569, … |
| maximal normal subgroup of | #72250548, #72250560, #72250569, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56395);