Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393889 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
988 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,11)(7,12,8,10)(13,14,16,17)(15,19,20,18)(21,22,26,29)(23,31,32,35)(24,30,25,28)(27,34,36,33);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(14,17)(18,19)(21,23,26,32)(22,27,29,36)(24,34,25,33)(28,35,30,31);(2,4)(5,8,9,7)(6,10,11,12)(14,17)(18,19)(22,28)(24,25)(27,35)(29,30)(31,36)(33,34);(13,15,16,20)(14,18,17,19)(22,29)(23,32)(28,30)(33,34)Polycyclic presentation:
2^9|a^2=efh,b^2=efi,c^2=ef,d^2=hi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=di,e^b=ei,f^b=fi,e^c=ei,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72308451, #72308459, #72308550, … |
| maximal normal subgroup of | #72308451, #72308459, #72308550, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 322647);