Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36424 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1873 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,24,27)(23,29,32,26)(25,28,33,35)(30,34,31,36);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,24,32)(22,26,27,29)(25,30,33,31)(28,36,35,34);(1,2,3,4)(22,27)(23,30)(25,33)(26,34)(29,36)(31,32);(6,11)(10,13)(12,19)(18,20);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(23,31)(25,33)(26,34)(28,35)(29,36)(30,32)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72284710, #72285043, #72307903, … |
| maximal normal subgroup of | #72284710, #72285043, #72307903, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593349);