Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15369 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1199 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)(3,8)(4,17,20,9)(5,22)(6,25)(7,11)(10,24)(12,29)(13,41,43,30)(14,44)(15,38)(16,32)(18,34,47,49)(19,35,53,50)(21,36,27,39)(23,37)(26,33)(28,40)(31,46)(42,56,64,63)(45,57)(48,58)(51,60,52,59)(54,62,55,61);(1,3,7,16)(2,8,11,32)(4,18,20,47)(5,14,24,46)(6,26,29,15)(9,34,17,49)(10,31,22,44)(12,38,25,33)(13,27,43,21)(19,52,53,51)(23,48,28,45)(30,39,41,36)(35,60,50,59)(37,58,40,57)(42,55,64,54)(56,62,63,61);(1,4)(2,9)(3,13)(5,19)(6,27)(7,20)(8,30)(10,35)(11,17)(12,39)(14,42)(15,47)(16,43)(18,26)(21,29)(22,50)(23,55)(24,53)(25,36)(28,54)(31,56)(32,41)(33,49)(34,38)(37,62)(40,61)(44,63)(45,51)(46,64)(48,52)(57,59)(58,60);(1,5)(2,10,11,22)(3,14)(4,19)(6,28)(7,24)(8,31,32,44)(9,35,17,50)(12,40,25,37)(13,42)(15,48)(16,46)(18,51)(20,53)(21,55)(23,29)(26,45)(27,54)(30,56,41,63)(33,58,38,57)(34,59,49,60)(36,62,39,61)(43,64)(47,52)Polycyclic presentation:
2^9|a^2=e,b^2=i,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,g^c=gi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334579);