Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140781 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1972 |
| 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 | |
724 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)(5,6,10,13,11,16,9,15)(7,17,20,34,22,28,21,36)(8,23,25,14,26,32,24,33)(12,27,29,19,30,18,31,35);(1,2,3,4)(5,7,8,19,10,21,25,27,11,22,26,35,9,20,24,18)(6,12,14,17,15,29,23,34,16,30,33,28,13,31,32,36);(6,13)(7,18)(8,24)(9,10)(12,28)(14,33)(15,16)(17,30)(19,20)(21,35)(22,27)(25,26)(29,34)(31,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,g^b=gi,d^c=dhi,e^c=ei,f^c=fhi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250148, #72250150, #72250157, … |
| maximal normal subgroup of | #72250148, #72250150, #72250157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55576);