Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459760 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16348 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,13,6,30)(3,14,17,9,19,34,18,33)(4,10,24,37,23,38,22,42)(5,25,29,11,28,40,27,41)(15,44,20,31,47,35,46,58)(16,48,50,43,52,32,51,60)(21,53,55,36,54,39,26,56)(45,62,49,59,64,57,63,61);(1,3,8,17,7,19,6,18)(2,9,12,34,13,33,30,14)(4,20,24,47,23,46,22,15)(5,16,29,50,28,52,27,51)(10,35,37,58,38,44,42,31)(11,32,40,60,41,48,25,43)(21,49,55,64,54,63,26,45)(36,61,39,62,56,59,53,57);(1,4,7,23)(2,10,13,38)(3,15,19,47)(5,26,28,55)(6,24,8,22)(9,31,33,58)(11,39,41,53)(12,42,30,37)(14,35,34,44)(16,49,52,63)(17,46,18,20)(21,29,54,27)(25,56,40,36)(32,59,48,61)(43,57,60,62)(45,51,64,50);(1,5,8,29,7,28,6,27)(2,11,12,40,13,41,30,25)(3,16,17,50,19,52,18,51)(4,21,24,55,23,54,22,26)(9,32,34,60,33,48,14,43)(10,36,37,39,38,56,42,53)(15,45,20,49,47,64,46,63)(31,57,35,61,58,62,44,59)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=i,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=de,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 366957);