Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498860 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#459 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,13)(9,15)(10,26)(11,16)(12,20)(14,23)(18,27)(19,34)(21,32)(22,33)(24,30)(25,31)(28,35)(29,36);(2,4)(5,7)(6,12)(8,22)(9,18)(10,21)(11,19)(13,17)(14,32)(15,33)(16,28)(20,23)(24,35)(25,36)(26,29)(27,30)(31,34);(6,13)(7,18)(8,23)(9,10)(12,27)(14,15)(16,31)(17,33)(19,21)(20,35)(24,25)(26,30)(28,32)(29,34);(7,19)(12,28)(17,34)(18,21)(20,35)(22,36)(27,32)(29,33)Polycyclic presentation:
2^9|e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,c^b=cg,d^b=di,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224928, #72224936, #72226810, … |
| maximal normal subgroup of | #72224928, #72224936, #72226810, … |
| maximal quotient of | #72224919 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406056);