Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806575 |
| 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 | #2011 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1744 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)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22,25,27)(23,28,31,35)(24,32,34,29)(26,30,33,36);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(22,27)(23,24)(26,33)(28,29)(31,34)(32,35);(1,2,3,4)(21,23,26,24,25,31,33,34)(22,28,30,32,27,35,36,29);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(22,27)(28,35)(29,32)(30,36);(7,15)(10,13)(14,20)(18,19)(21,24,33,23,25,34,26,31)(22,29,36,35,27,32,30,28)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=fi,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72308353 |
| maximal normal subgroup of | #72308353 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722423);