Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #390446 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14572 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1318 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)(4,18,22,40)(5,23)(6,12)(7,11)(9,21,34,27)(10,24)(14,42)(15,45,47,57)(16,39,26,32)(19,51)(20,53,55,33)(25,37)(28,36)(29,38)(30,44)(31,46,35,49)(41,48)(43,59)(50,56,58,54)(52,60)(61,62)(63,64);(1,3)(2,8)(4,19)(5,15)(6,26)(7,17)(9,30)(10,35)(11,13)(12,39)(14,22)(16,29)(18,42)(20,52)(21,44)(23,57)(24,49)(25,47)(27,48)(28,46)(31,36)(32,38)(33,60)(34,41)(37,45)(40,51)(43,55)(50,64)(53,59)(54,62)(56,63)(58,61);(1,4)(2,9)(3,14,17,19)(5,20)(6,27)(7,22)(8,30,13,41)(10,33)(11,34)(12,40)(15,43,47,52)(16,48,26,44)(18,38)(21,29)(23,58)(24,56)(25,55)(28,54)(31,59,35,60)(32,42,39,51)(36,53)(37,50)(45,64,57,61)(46,63,49,62);(1,5,7,25)(2,10,11,36)(3,15,17,47)(4,20,22,55)(6,28,29,24)(8,31,13,35)(9,33,34,53)(12,23,38,37)(14,43,19,52)(16,49,26,46)(18,50,40,58)(21,56,27,54)(30,59,41,60)(32,57,39,45)(42,61,51,64)(44,63,48,62)Polycyclic presentation:
2^9|a^2=h,c^2=egh,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 297643);