Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #401086 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14811 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,8,15,30)(4,9)(5,21)(6,24)(7,11)(10,23)(12,27)(13,39,41,28)(14,42,43,29)(16,37,25,31)(17,32,45,56)(18,47)(19,38)(20,34)(22,36)(26,35)(33,49)(40,54,62,61)(44,55,51,59)(46,64,63,57)(48,58)(50,53)(52,60);(1,3)(2,8,11,30)(4,17)(5,14)(6,25)(7,15)(9,32,34,56)(10,29,21,42)(12,37,24,31)(13,19)(16,27)(18,46)(20,45)(22,51)(23,43)(26,41)(28,38,39,35)(33,57,47,64)(36,59,50,55)(40,48)(44,53)(49,63)(52,62)(54,60,61,58);(1,4,7,20)(2,9,11,34)(3,13,15,41)(5,18,23,49)(6,26,27,19)(8,28,30,39)(10,33,21,47)(12,38,24,35)(14,40,43,62)(16,45,25,17)(22,52,53,48)(29,54,42,61)(31,56,37,32)(36,60,50,58)(44,63,51,46)(55,64,59,57);(1,5)(2,10,11,21)(3,14)(4,18)(6,22)(7,23)(8,29,30,42)(9,33,34,47)(12,36,24,50)(13,40)(15,43)(16,44)(17,46)(19,48)(20,49)(25,51)(26,52)(27,53)(28,54,39,61)(31,55,37,59)(32,57,56,64)(35,58,38,60)(41,62)(45,63)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=i,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308283);