Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394821 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14719 |
| 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 | |
1528 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)(4,21)(5,27)(6,13)(7,12)(8,14)(9,18)(10,25)(11,28)(16,35)(17,46)(19,34)(20,33)(22,51)(23,53)(24,37)(26,38)(29,39)(30,40)(31,45)(32,49)(36,56)(41,44)(42,47)(43,58)(48,59)(50,54)(52,60)(55,57)(61,63)(62,64);(1,3,6,18,8,20,7,19)(2,9,12,15,14,34,13,33)(4,22,24,44,25,45,26,16)(5,17,28,47,29,49,30,48)(10,35,37,31,21,41,38,51)(11,32,27,42,40,46,39,59)(23,52,54,62,55,63,56,43)(36,60,57,58,50,61,53,64);(1,4,6,24,8,25,7,26)(2,10,12,37,14,21,13,38)(3,16,18,22,20,44,19,45)(5,23,28,54,29,55,30,56)(9,31,15,41,34,51,33,35)(11,36,27,57,40,50,39,53)(17,43,47,52,49,62,48,63)(32,58,42,61,46,64,59,60);(1,5)(2,11)(3,17)(4,23)(6,28)(7,30)(8,29)(9,32)(10,36)(12,27)(13,39)(14,40)(15,42)(16,43)(18,47)(19,48)(20,49)(21,50)(22,52)(24,54)(25,55)(26,56)(31,58)(33,59)(34,46)(35,60)(37,57)(38,53)(41,61)(44,62)(45,63)(51,64)Polycyclic presentation:
2^9|b^2=e,c^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302017);