Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23998 |
| 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 | |
1225 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,25)(3,9,18,30,16,31,17,39)(4,19,21,10,22,35,26,34)(5,11,24,38,23,37,27,36)(14,40,43,28,42,54,47,32)(15,44,45,55,46,29,48,56)(20,49,52,58,50,59,51,33)(41,61,63,53,64,60,62,57);(1,3,7,17,8,16,6,18)(2,9,25,30,12,31,13,39)(4,14,21,47,22,42,26,43)(5,15,27,45,23,46,24,48)(10,32,19,40,34,28,35,54)(11,29,38,55,37,44,36,56)(20,41,51,63,50,64,52,62)(33,57,49,60,58,53,59,61);(1,4)(2,10)(3,14)(5,20)(6,26)(7,21)(8,22)(9,28)(11,33)(12,34)(13,35)(15,41)(16,42)(17,47)(18,43)(19,25)(23,50)(24,52)(27,51)(29,53)(30,54)(31,32)(36,59)(37,58)(38,49)(39,40)(44,57)(45,63)(46,64)(48,62)(55,61)(56,60);(1,5)(2,11,12,37)(3,15)(4,20)(6,24)(7,27)(8,23)(9,29,31,44)(10,33,34,58)(13,36,25,38)(14,41)(16,46)(17,45)(18,48)(19,49,35,59)(21,51)(22,50)(26,52)(28,53,32,57)(30,55,39,56)(40,60,54,61)(42,64)(43,62)(47,63)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,f^c=fi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 278221);