Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #368859 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14023 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1635 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)(3,14)(4,20,24,49)(5,25)(6,28)(7,12)(8,13)(9,17)(10,23,35,22)(11,26)(15,33,44,30)(16,31)(18,32)(19,38)(21,34,51,59)(27,36)(29,37)(39,43,58,42)(40,45)(41,60,62,64)(46,56)(47,55)(48,52,53,50)(54,61,57,63);(1,3)(2,9)(4,15,24,44)(5,16)(6,18)(7,17)(8,19)(10,33,35,30)(11,31)(12,38)(13,32)(14,28)(20,39,49,58)(21,41,51,62)(22,43,23,42)(25,40)(26,47)(27,46)(29,45)(34,57,59,54)(36,55)(37,56)(48,60,53,64)(50,63,52,61);(1,4,6,22,8,24,7,23)(2,10,12,20,13,35,28,49)(3,15,18,43,19,44,17,42)(5,21,29,52,27,51,26,50)(9,30,38,58,32,33,14,39)(11,34,36,48,37,59,25,53)(16,41,45,61,46,62,47,63)(31,54,55,64,56,57,40,60);(1,5,8,27)(2,11,13,37)(3,16,19,46)(4,21,24,51)(6,29,7,26)(9,31,32,56)(10,34,35,59)(12,36,28,25)(14,40,38,55)(15,41,44,62)(17,47,18,45)(20,48,49,53)(22,52,23,50)(30,54,33,57)(39,60,58,64)(42,63,43,61)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276056);