Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137661 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| 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 | |
660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,16,18,11)(8,12)(9,25)(10,13)(14,23)(15,22)(17,29,20,26)(19,28,35,33)(21,31)(24,30)(27,36,32,34);(1,2)(3,4)(5,7)(6,11)(8,20)(9,19)(10,18)(12,29)(13,16)(14,32)(15,28)(17,24)(21,36)(22,35)(23,34)(25,33)(26,30)(27,31);(2,4)(6,12,13,30)(8,21)(9,22)(11,26,16,29)(14,25,31,15)(17,34)(19,35)(20,36)(23,24)(27,33,32,28)Polycyclic presentation:
2^9|a^2=g,c^2=d,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^b=gi,e^c=ehi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249872, #72249873, #72249882, … |
| maximal normal subgroup of | #72249872, #72249873, #72249882, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55805);