Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #428071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2504 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
857 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6,12,16,11,17,10,30)(7,18,21,13,22,42,31,36)(8,14,24,53,25,39,32,38)(9,26,33,41,29,55,28,15)(19,43,54,59,46,62,45,34)(20,47,27,40,48,35,56,63)(23,49,51,37,52,66,57,61)(44,64,67,68,65,58,50,60);(5,7,12,21,11,22,10,31)(6,13,16,42,17,36,30,18)(8,19,24,54,25,46,32,45)(9,27,33,48,29,56,28,20)(14,34,53,43,39,59,38,62)(15,40,26,35,41,63,55,47)(23,50,51,44,52,67,57,65)(37,60,66,64,61,68,49,58);(5,8,10,24,11,25,12,32)(6,14,30,53,17,39,16,38)(7,19,21,45,22,46,31,54)(9,23,28,51,29,52,33,57)(13,34,42,62,36,59,18,43)(15,37,55,66,41,61,26,49)(20,44,27,50,48,65,56,67)(35,58,63,68,47,64,40,60);(2,4)(5,9,12,33,11,29,10,28)(6,15,30,55,17,41,16,26)(7,20,31,56,22,48,21,27)(8,23,32,57,25,52,24,51)(13,35,42,63,36,47,18,40)(14,37,53,66,39,61,38,49)(19,44,54,67,46,65,45,50)(34,58,62,68,59,64,43,60)Polycyclic presentation:
2^9|a^2=efg,b^2=efg,c^2=efi,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,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, 349330);