Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24985 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2420 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,19)(8,23)(9,13)(10,17)(11,16)(12,18)(14,26)(15,22)(20,35)(21,30)(24,32)(25,31)(27,33)(28,34)(29,36);(1,3,4,2)(5,7,11,21,12,22,10,9)(6,13,17,15,18,30,16,19)(8,24,26,20,28,25,27,36)(14,31,23,35,33,32,34,29);(5,8,10,26,12,28,11,27)(6,14,17,34,18,33,16,23)(7,20,9,25,22,36,21,24)(13,29,15,32,30,35,19,31);(5,9,10,22,12,21,11,7)(6,15,16,13,18,19,17,30)(8,25,26,36,28,24,27,20)(14,32,23,29,33,31,34,35)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=efg,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276745);