Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #381483 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14299 |
| 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 | |
1337 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)(4,21,26,10)(5,11)(6,14)(7,12)(8,13)(9,18)(16,44,47,35)(17,43)(19,33)(20,34)(22,53,46,31)(23,52,55,36)(24,37,25,38)(27,39,56,63)(28,42)(29,41)(30,40)(32,49)(45,59,57,62)(48,58,54,64)(50,61)(51,60);(1,3,8,20)(2,9,13,33)(4,22,26,46)(5,17,29,50)(6,19,7,18)(10,35,21,44)(11,32,41,60)(12,34,14,15)(16,24,47,25)(23,54,55,48)(27,57,56,45)(28,51,30,49)(31,37,53,38)(36,62,52,59)(39,64,63,58)(40,61,42,43);(1,4)(2,10)(3,16)(5,27)(6,25)(7,24)(8,26)(9,31)(11,39)(12,38)(13,21)(14,37)(15,35)(17,48)(18,22)(19,46)(20,47)(23,28)(29,56)(30,55)(32,59)(33,53)(34,44)(36,40)(41,63)(42,52)(43,58)(45,49)(50,54)(51,57)(60,62)(61,64);(1,5)(2,11)(3,17)(4,23,26,55)(6,28)(7,30)(8,29)(9,32)(10,36,21,52)(12,40)(13,41)(14,42)(15,43)(16,45,47,57)(18,49)(19,51)(20,50)(22,54,46,48)(24,27,25,56)(31,58,53,64)(33,60)(34,61)(35,62,44,59)(37,39,38,63)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^c=hi,f^d=fi,g^d=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288680);