Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2749 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14507 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
751 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)(3,13,17,8)(4,18,20,37)(5,22)(6,12)(7,11)(9,21,33,26)(10,34)(14,40)(15,44,23,51)(16,39,47,30)(19,32,50,60)(24,53)(25,35)(27,36)(28,43)(29,55,38,45)(31,46)(41,59)(42,58)(48,56,52,49)(54,57)(61,62)(63,64);(1,3,6,16,7,17,27,47)(2,8,12,30,11,13,36,39)(4,14,21,43,20,42,26,46)(5,23,25,55,24,15,34,45)(9,31,18,40,33,28,37,58)(10,29,22,51,35,38,53,44)(19,41,49,62,50,54,56,64)(32,57,48,63,60,59,52,61);(1,4)(2,9)(3,14,17,42)(5,19)(6,26)(7,20)(8,28,13,31)(10,32)(11,33)(12,37)(15,41,23,54)(16,46,47,43)(18,36)(21,27)(22,52)(24,50)(25,56)(29,57,38,59)(30,40,39,58)(34,49)(35,60)(44,63,51,61)(45,64,55,62)(48,53);(1,5)(2,10)(3,15,17,23)(4,19)(6,25)(7,24)(8,29,13,38)(9,32)(11,35)(12,22)(14,41,42,54)(16,45,47,55)(18,48)(20,50)(21,49)(26,56)(27,34)(28,57,31,59)(30,51,39,44)(33,60)(36,53)(37,52)(40,61,58,63)(43,62,46,64)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=eg,d^2=eg,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295200);