Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395822 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14727 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1736 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)(3,15,19,33)(4,10)(5,24,26,40)(6,12)(7,13)(8,14)(9,18,32,27)(11,25,39,29)(16,34,44,41)(17,31)(20,30,45,57)(21,50,52,55)(22,36)(23,38)(28,37)(35,53,60,51)(42,46)(43,56)(47,54)(48,58)(49,61)(59,63)(62,64);(1,3)(2,9)(4,20)(5,17)(6,27)(7,18)(8,19)(10,34)(11,31)(12,33)(13,15)(14,32)(16,22)(21,49)(23,45)(24,54)(25,47)(26,48)(28,44)(29,46)(30,37)(35,59)(36,57)(38,41)(39,58)(40,42)(43,51)(50,56)(52,63)(53,62)(55,64)(60,61);(1,4,6,22,8,23,7,28)(2,10,12,36,14,38,13,37)(3,16,18,20,19,44,27,45)(5,21,25,51,26,52,29,53)(9,30,15,41,32,57,33,34)(11,35,24,55,39,60,40,50)(17,43,46,49,48,62,47,63)(31,56,42,61,58,64,54,59);(1,5,8,26)(2,11,14,39)(3,17,19,48)(4,21,23,52)(6,25,7,29)(9,31,32,58)(10,35,38,60)(12,24,13,40)(15,42,33,54)(16,43,44,62)(18,46,27,47)(20,49,45,63)(22,51,28,53)(30,56,57,64)(34,59,41,61)(36,55,37,50)Polycyclic presentation:
2^9|a^2=f,c^2=e,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^c=hi,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72287280, #72287292, #72287345, … |
| maximal normal subgroup of | #72287280, #72287292, #72287345, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303019);