Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682420 |
| 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 | #2416 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22)(23,30)(24,32)(25,29)(26,27)(28,33)(31,36)(34,35);(1,2)(3,4)(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,25,26)(22,27,29,30)(24,33,34,31)(28,32,36,35);(2,4)(21,24,26,33,25,34,23,31)(22,28,30,32,29,36,27,35);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(22,29)(24,34)(27,30)(31,33);(2,4)(7,15)(10,13)(14,20)(18,19)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=g,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307811, #72307977, #72308554 |
| maximal normal subgroup of | #72307811, #72307977, #72308554 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591244);