Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135286 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1823 |
| 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 | |
650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,16,11,14,10,26)(7,17,19,30,20,12,25,29)(8,21,23,31,24,13,22,15)(18,27,34,33,35,32,36,28);(1,2)(3,4)(5,7)(6,12)(8,18)(9,25)(10,19)(11,20)(13,27)(14,17)(15,28)(16,30)(21,32)(22,34)(23,36)(24,35)(26,29)(31,33);(2,4)(6,13,14,21)(8,22)(9,10)(12,27,17,32)(15,16,31,26)(18,34)(19,25)(23,24)(28,30,33,29)(35,36)Polycyclic presentation:
2^9|a^2=g,c^2=d,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,c^b=cf,g^b=gi,e^c=eh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249860, #72249878, #72249879, … |
| maximal normal subgroup of | #72249860, #72249878, #72249879, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55800);