Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1005 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)(7,17,21,27)(8,22)(9,12)(10,15)(11,16)(13,18,28,20)(14,24)(19,33,34,26)(23,30)(25,29)(31,32,36,35);(6,12)(7,18)(8,23)(9,10)(15,16)(17,27)(19,34)(20,21)(22,29)(24,25)(26,36)(31,33);(5,7,11,21)(6,13)(8,19,25,34)(9,18,10,20)(12,17)(14,26)(15,27)(16,28)(22,31)(23,32,24,35)(29,36)(30,33);(2,4)(5,8,10,24,11,25,9,23)(6,14,12,22,16,30,15,29)(7,19,20,35,21,34,18,32)(13,26,17,31,28,33,27,36)Polycyclic presentation:
2^9|a^2=eg,c^2=ef,d^2=ei,e^2=i,f^2=i,g^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,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341665);