Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395163 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| 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 | |
1098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(3,7)(5,8)(9,10,15,20,16,21,14,36)(11,22,26,40,27,47,25,17)(12,28,30,43,31,42,37,18)(13,32,34,63,35,19,33,44)(23,38,49,56,50,66,55,45)(24,51,53,39,54,46,52,61)(29,58,59,41,60,57,64,62)(48,68,69,67,70,65,72,71);(1,2,4,6)(3,8,7,5)(9,11,15,26,16,27,14,25)(10,17,36,47,21,40,20,22)(12,23,37,55,31,50,30,49)(13,24,34,53,35,54,33,52)(18,38,28,56,43,66,42,45)(19,39,63,51,32,61,44,46)(29,48,64,72,60,70,59,69)(41,65,57,71,62,68,58,67);(1,3,4,7)(2,5,6,8)(9,12,16,31)(10,18,21,43)(11,23,27,50)(13,29,35,60)(14,30,15,37)(17,38,40,66)(19,41,32,62)(20,42,36,28)(22,45,47,56)(24,48,54,70)(25,49,26,55)(33,59,34,64)(39,65,61,68)(44,58,63,57)(46,67,51,71)(52,69,53,72);(9,13,14,33,16,35,15,34)(10,19,20,44,21,32,36,63)(11,24,25,52,27,54,26,53)(12,29,30,59,31,60,37,64)(17,39,22,46,40,61,47,51)(18,41,42,58,43,62,28,57)(23,48,49,69,50,70,55,72)(38,65,45,67,66,68,56,71)Polycyclic presentation:
2^9|a^2=fg,b^2=egi,c^2=gi,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311058);