Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #462060 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16415 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
716 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,26,8,12,7,13)(3,9,17,33,16,31,27,32)(4,10,20,38,21,36,28,37)(5,22,23,11,25,41,24,40)(14,42,18,34,44,29,45,59)(15,46,49,57,48,58,50,30)(19,52,53,39,54,61,55,35)(43,60,64,63,47,62,51,56);(1,3)(2,9,12,31)(4,18)(5,15)(6,27)(7,17)(8,16)(10,34,36,59)(11,30,40,57)(13,33,26,32)(14,20)(19,51)(21,45)(22,46,41,58)(23,50)(24,49)(25,48)(28,44)(29,38,42,37)(35,60,39,62)(43,55)(47,53)(52,56,61,63)(54,64);(1,4)(2,10)(3,14,16,44)(5,19)(6,20)(7,28)(8,21)(9,29,31,42)(11,39)(12,36)(13,37)(15,47,48,43)(17,18,27,45)(22,52)(23,53)(24,55)(25,54)(26,38)(30,56,57,63)(32,34,33,59)(35,40)(41,61)(46,60,58,62)(49,51,50,64);(1,5,8,25)(2,11,12,40)(3,15,16,48)(4,19,21,54)(6,24,7,23)(9,30,31,57)(10,35,36,39)(13,41,26,22)(14,43,44,47)(17,50,27,49)(18,51,45,64)(20,55,28,53)(29,56,42,63)(32,46,33,58)(34,60,59,62)(37,52,38,61)Polycyclic presentation:
2^9|a^2=efg,b^2=ef,c^2=f,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 369257);