Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135930 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1864 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
754 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)(5,6,8,16,11,17,24,14)(7,18,20,12,23,33,22,34)(9,26,25,15,28,13,10,29)(19,31,21,36,30,35,27,32);(5,7,8,20,11,23,24,22)(6,12,14,18,17,34,16,33)(9,27,25,19,28,21,10,30)(13,35,15,36,26,31,29,32);(2,4)(6,13)(7,19)(8,24)(9,10)(12,18)(14,29)(15,16)(17,26)(20,27)(21,22)(23,30)(25,28)(31,35)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=eh,f^b=fhi,h^b=hi,d^c=di,e^c=ehi,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249861, #72249871, #72249881, … |
| maximal normal subgroup of | #72249861, #72249871, #72249881, … |
| maximal quotient of | #72265623, C2×#148152 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55348);