Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #397952 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14757 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1810 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)(3,15)(4,10)(5,26)(6,14)(7,12)(8,13)(9,18)(11,27)(16,42)(17,46)(19,33)(20,32)(21,40)(22,51)(23,36)(24,37)(25,35)(28,39)(29,38)(30,45)(31,48)(34,52)(41,47)(43,55)(44,56)(49,57)(50,64)(53,59)(54,58)(60,63)(61,62);(1,3,6,18,8,19,7,20)(2,9,12,15,13,32,14,33)(4,21,23,45,24,16,25,44)(5,17,27,47,28,48,29,49)(10,30,35,40,37,56,36,42)(11,31,26,41,38,46,39,57)(22,50,52,63,53,43,54,62)(34,55,51,60,58,64,59,61);(1,4,8,24)(2,10,13,37)(3,16)(5,22,28,53)(6,23,7,25)(9,30)(11,34,38,58)(12,35,14,36)(15,40)(17,43)(18,44)(19,21)(20,45)(26,51,39,59)(27,52,29,54)(31,55)(32,56)(33,42)(41,60)(46,64)(47,62)(48,50)(49,63)(57,61);(1,5)(2,11)(3,17)(4,22)(6,27)(7,29)(8,28)(9,31)(10,34)(12,26)(13,38)(14,39)(15,41)(16,43)(18,47)(19,48)(20,49)(21,50)(23,52)(24,53)(25,54)(30,55)(32,46)(33,57)(35,51)(36,59)(37,58)(40,60)(42,61)(44,62)(45,63)(56,64)Polycyclic presentation:
2^9|b^2=e,c^2=fi,e^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,f^b=fi,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, 305149);