Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395174 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2404 |
| prime radical | #23406 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,12)(8,20)(9,16)(10,14)(11,15)(13,22)(17,32)(18,26)(19,28)(21,31)(23,27)(24,30)(25,34)(29,35)(33,36);(1,2,3,4)(5,7)(6,12)(8,17)(9,23)(10,18)(11,19)(13,29)(14,26)(15,28)(16,27)(20,32)(21,33)(22,35)(24,34)(25,30)(31,36);(5,8)(6,13)(7,17)(9,24)(10,22)(11,21)(12,25)(14,31)(15,30)(16,20)(18,35)(19,33)(23,34)(26,32)(27,36)(28,29);(6,14,15,16)(8,21)(12,26,28,27)(13,20,30,31)(17,33)(22,24)(25,36,29,32)(34,35)Polycyclic presentation:
2^9|b^2=e,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280086 |
| maximal normal subgroup of | #72280086 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311274);