Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418498 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2673 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15412 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1491 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)(3,9,17,31)(4,10)(5,23)(6,14)(7,13)(8,12)(11,25)(15,42,44,29)(16,46,47,55)(18,39,26,32)(19,33,45,57)(20,51)(21,40)(22,35)(24,41)(27,36)(28,38)(30,48,37,49)(34,54)(43,59,63,58)(50,62,64,56)(52,60)(53,61);(1,3)(2,9,12,31)(4,19)(5,16,24,47)(6,26)(7,18)(8,17)(10,33,35,57)(11,37)(13,39,14,32)(15,21)(20,50,52,64)(22,45)(23,55)(25,49,28,48)(27,44)(29,40,42,36)(30,38)(34,59)(41,46)(43,54,63,53)(51,56)(58,61)(60,62);(1,4)(2,10)(3,15)(5,20)(6,27)(7,21)(8,22)(9,29)(11,34)(12,35)(13,40)(14,36)(16,43)(17,44)(18,45)(19,26)(23,51)(24,52)(25,54)(28,53)(30,56)(31,42)(32,57)(33,39)(37,62)(38,61)(41,60)(46,58)(47,63)(48,64)(49,50)(55,59);(1,5)(2,11)(3,16,17,47)(4,20)(6,25)(7,28)(8,24)(9,30,31,37)(10,34)(12,38)(13,23)(14,41)(15,43,44,63)(18,48,26,49)(19,50,45,64)(21,53)(22,52)(27,54)(29,56,42,62)(32,55,39,46)(33,58,57,59)(35,61)(36,60)(40,51)Polycyclic presentation:
2^9|a^2=f,b^2=ef,d^2=f,g^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,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi,e^d=ei,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337270);