Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455836 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16270 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12)(3,13,17,37)(4,18,21,9)(5,10,24,36)(6,25,28,11)(8,26,31,16)(14,41,44,39)(15,45,48,40)(19,32,53,51)(20,33,56,52)(22,57,55,34)(23,38,27,35)(29,43,50,54)(30,49,61,47)(42,60,63,62)(46,59,58,64);(1,3)(2,8)(4,14,21,44)(5,15)(6,26)(7,17)(9,29,18,50)(10,30)(11,37)(12,31)(13,25)(16,28)(19,42,53,63)(20,54,56,43)(22,58,55,46)(23,49)(24,48)(27,47)(32,59,51,64)(33,41,52,39)(34,62,57,60)(35,40)(36,61)(38,45);(1,4,6,20,7,21,28,56)(2,9,11,33,12,18,25,52)(3,14,16,43,17,44,26,54)(5,22,23,53,24,55,27,19)(8,29,13,39,31,50,37,41)(10,34,35,51,36,57,38,32)(15,46,47,63,48,58,49,42)(30,60,45,64,61,62,40,59);(1,5)(2,10)(3,15)(4,19)(6,27)(7,24)(8,30)(9,32)(11,38)(12,36)(13,40)(14,42)(16,49)(17,48)(18,51)(20,55)(21,53)(22,56)(23,28)(25,35)(26,47)(29,59)(31,61)(33,57)(34,52)(37,45)(39,62)(41,60)(43,58)(44,63)(46,54)(50,64)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363033);