Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419010 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15267 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2194 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)(3,15)(4,10)(5,26,29,41)(6,13)(7,12)(8,14)(9,18)(11,30,40,28)(16,43)(17,32,47,55)(19,33)(20,42)(21,31)(22,51,53,56)(23,37)(24,36)(25,38)(27,44,48,39)(34,46)(35,52,62,54)(45,61,64,58)(49,59)(50,63,57,60);(1,3,6,18,8,20,7,19)(2,9,12,15,14,33,13,42)(4,21,23,46,25,49,24,16)(5,27,28,47,29,48,30,17)(10,34,36,31,38,43,37,59)(11,39,26,55,40,44,41,32)(22,45,52,57,53,64,54,50)(35,61,56,63,62,58,51,60);(1,4,8,25)(2,10,14,38)(3,16,20,46)(5,22,29,53)(6,23,7,24)(9,31,33,59)(11,35,40,62)(12,36,13,37)(15,43,42,34)(17,45,47,64)(18,21,19,49)(26,56,41,51)(27,57,48,50)(28,52,30,54)(32,58,55,61)(39,60,44,63);(1,5)(2,11)(3,17)(4,22)(6,30)(7,28)(8,29)(9,32)(10,35)(12,41)(13,26)(14,40)(15,44)(16,45)(18,48)(19,27)(20,47)(21,50)(23,54)(24,52)(25,53)(31,58)(33,55)(34,60)(36,51)(37,56)(38,62)(39,42)(43,63)(46,64)(49,57)(59,61)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72286957, #72287270, #72287302, … |
| maximal normal subgroup of | #72286957, #72287270, #72287302, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326207);