Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395174 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,13)(9,16)(10,14)(11,15)(12,19)(18,33)(20,27)(21,26)(22,31)(23,29)(24,30)(25,36)(28,35)(32,34);(2,4)(5,7,10,21,11,20,9,19)(6,12,16,27,15,26,14,17)(8,18,24,36,22,34,23,35)(13,28,29,32,31,25,30,33);(5,8)(6,13)(7,18)(9,23)(10,24)(11,22)(12,25)(14,30)(15,31)(16,29)(17,32)(19,35)(20,34)(21,36)(26,28)(27,33);(6,14,15,16)(8,22)(12,17,26,27)(13,29,31,30)(18,34)(23,24)(25,33,28,32)(35,36)Polycyclic presentation:
2^9|b^2=h,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250788, #72250789, #72250791, … |
| maximal normal subgroup of | #72250788, #72250789, #72250791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318508);