Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23990 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2404 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14064 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1227 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)(3,14)(4,20,24,38)(5,25)(6,28)(7,12)(8,13)(9,17)(10,23,35,22)(11,26)(15,30,45,33)(16,46)(18,32)(19,41)(21,51,53,34)(27,36)(29,37)(31,48)(39,44,57,43)(40,47)(42,61,63,60)(49,58)(50,54,55,52)(56,64,59,62);(1,3,6,17,8,19,7,18)(2,9,12,14,13,32,28,41)(4,15,23,44,24,45,22,43)(5,16,26,47,27,48,29,49)(10,33,38,57,35,30,20,39)(11,31,25,40,37,46,36,58)(21,42,54,64,53,63,52,62)(34,59,55,61,51,56,50,60);(1,4)(2,10)(3,15)(5,21)(6,23)(7,22)(8,24)(9,30)(11,34)(12,38)(13,35)(14,39)(16,42)(17,44)(18,43)(19,45)(20,28)(25,55)(26,54)(27,53)(29,52)(31,56)(32,33)(36,50)(37,51)(40,60)(41,57)(46,59)(47,64)(48,63)(49,62)(58,61);(1,5)(2,11,13,37)(3,16)(4,21)(6,26)(7,29)(8,27)(9,31,32,46)(10,34,35,51)(12,25,28,36)(14,40,41,58)(15,42)(17,47)(18,49)(19,48)(20,50,38,55)(22,52)(23,54)(24,53)(30,56,33,59)(39,60,57,61)(43,62)(44,64)(45,63)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^c=ei,f^c=fi,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 278210);