Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24805 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#121 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,17,20,28)(8,21)(9,12)(10,15)(11,16)(13,19,29,25)(14,23)(18,27,32,36)(22,26)(24,30)(31,35,34,33);(2,4)(6,12,16,15)(8,22)(13,17,29,28)(14,21,30,26)(18,32)(23,24)(27,34,36,31)(33,35);(1,2,3,4)(5,7)(6,13)(8,18)(9,19)(10,25)(11,20)(12,17)(14,27)(15,28)(16,29)(21,34)(22,32)(23,33)(24,35)(26,31)(30,36);(5,8,9,23,11,22,10,24)(6,14,15,21,16,30,12,26)(7,18,19,33,20,32,25,35)(13,27,28,34,29,36,17,31)Polycyclic presentation:
2^9|a^2=h,b^2=efh,c^2=g,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250001, #72250013, #72250427, … |
| maximal normal subgroup of | #72250001, #72250013, #72250427, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275856);