Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #391902 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2381 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14670 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
551 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,14,7,13)(3,15,18,33,19,43,29,9)(4,10,23,37,24,38,22,36)(5,11,27,41,28,42,30,44)(16,46,20,31,49,34,48,58)(17,45,51,60,52,39,25,32)(21,35,55,63,56,64,26,40)(47,59,53,57,54,61,50,62);(1,3,7,18,8,19,6,29)(2,9,13,15,14,33,12,43)(4,20,22,49,24,48,23,16)(5,25,27,52,28,51,30,17)(10,34,36,58,38,46,37,31)(11,39,41,60,42,45,44,32)(21,54,55,53,56,47,26,50)(35,62,63,61,64,57,40,59);(1,4)(2,10)(3,16,19,49)(5,26)(6,23)(7,22)(8,24)(9,31,33,58)(11,40)(12,37)(13,36)(14,38)(15,34,43,46)(17,50,52,53)(18,20,29,48)(21,27)(25,54,51,47)(28,55)(30,56)(32,59,60,61)(35,41)(39,62,45,57)(42,63)(44,64);(1,5,6,27,8,28,7,30)(2,11,12,41,14,42,13,44)(3,17,18,51,19,52,29,25)(4,21,22,26,24,56,23,55)(9,32,15,45,33,60,43,39)(10,35,36,40,38,64,37,63)(16,47,48,50,49,54,20,53)(31,57,46,59,58,62,34,61)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,d^c=de,e^c=ei,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299197);