Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,10,13)(7,14,16,11)(8,17,9,12)(15,19,18,20);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,22,25,28)(23,29,30,36)(24,33,31,27)(26,34,35,32);(2,4)(21,23,24,32,25,30,31,34)(22,26,27,29,28,35,33,36);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(23,30)(26,35)(29,36)(32,34);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,23,31,34,25,30,24,32)(22,26,33,36,28,35,27,29)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=h,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72308009 |
| maximal normal subgroup of | #72308009 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6719128);