Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #380391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2331 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14266 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
673 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)(3,14,19,42)(4,20)(5,11)(6,12)(7,29)(8,13)(9,17,32,18)(10,22)(15,33,45,30)(16,41,49,62)(21,52)(23,35)(24,34)(25,38)(26,55)(27,39)(28,36)(31,47,58,48)(37,53)(40,50,51,44)(43,59,63,56)(46,61,64,57)(54,60);(1,3,7,18,8,19,6,17)(2,9,12,14,13,32,29,42)(4,15,22,44,23,45,28,50)(5,16,26,48,27,49,25,47)(10,33,20,51,36,30,35,40)(11,31,38,41,39,58,55,62)(21,43,24,46,53,63,54,64)(34,59,52,57,60,56,37,61);(1,4,7,22,8,23,6,28)(2,10,12,20,13,36,29,35)(3,15,18,44,19,45,17,50)(5,24,26,53,27,54,25,21)(9,30,14,40,32,33,42,51)(11,37,38,34,39,52,55,60)(16,46,48,63,49,64,47,43)(31,57,41,56,58,61,62,59);(1,5,6,25,8,27,7,26)(2,11,12,38,13,39,29,55)(3,16,17,47,19,49,18,48)(4,21,22,24,23,53,28,54)(9,31,14,41,32,58,42,62)(10,34,35,37,36,60,20,52)(15,43,44,46,45,63,50,64)(30,56,51,57,33,59,40,61)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287587);