Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418871 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2697 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15328 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
995 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,6,13,7,26)(3,9,27,32,16,33,17,39)(4,18,21,10,22,40,28,36)(5,11,29,37,24,38,25,53)(14,41,43,30,19,48,46,34)(15,44,52,59,23,31,45,56)(20,47,50,35,51,60,54,58)(42,62,63,57,49,61,64,55);(1,3)(2,9)(4,19)(5,23)(6,16)(7,27)(8,17)(10,34)(11,31)(12,39)(13,33)(14,22)(15,24)(18,41)(20,49)(21,43)(25,45)(26,32)(28,46)(29,52)(30,36)(35,55)(37,59)(38,44)(40,48)(42,51)(47,62)(50,63)(53,56)(54,64)(57,58)(60,61);(1,4,8,21,6,22,7,28)(2,10,12,40,13,36,26,18)(3,14,27,43,16,19,17,46)(5,20,29,50,24,51,25,54)(9,30,32,48,33,34,39,41)(11,35,37,60,38,58,53,47)(15,42,52,63,23,49,45,64)(31,55,56,62,44,57,59,61);(1,5)(2,11)(3,15)(4,20)(6,24)(7,25)(8,29)(9,31)(10,35)(12,37)(13,38)(14,42)(16,23)(17,45)(18,47)(19,49)(21,50)(22,51)(26,53)(27,52)(28,54)(30,55)(32,56)(33,44)(34,57)(36,58)(39,59)(40,60)(41,61)(43,63)(46,64)(48,62)Polycyclic presentation:
2^9|a^2=ef,c^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 332286);