Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #464193 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2657 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16809 |
| prime radical | #23414 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ 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)(3,14)(4,10)(5,24,27,55)(6,29)(7,13)(8,12)(9,17)(11,28,39,26)(15,40)(16,45,48,32)(18,42)(19,33)(20,51)(21,53,54,38)(22,36)(23,37)(25,56,58,34)(30,35)(31,50)(41,49,63,47)(43,64,52,59)(44,60)(46,61,57,62);(1,3,6,17,7,18,8,19)(2,9,12,14,13,33,29,42)(4,20,30,44,23,15,22,50)(5,16,26,47,27,48,28,49)(10,31,36,40,37,60,35,51)(11,32,24,41,39,45,55,63)(21,52,58,46,54,43,25,57)(34,59,53,62,56,64,38,61);(1,4)(2,10)(3,15)(5,25)(6,30)(7,23)(8,22)(9,31)(11,38)(12,36)(13,37)(14,40)(16,46)(17,50)(18,20)(19,44)(21,26)(24,34)(27,58)(28,54)(29,35)(32,61)(33,60)(39,53)(41,59)(42,51)(43,47)(45,62)(48,57)(49,52)(55,56)(63,64);(1,5,6,26,7,27,8,28)(2,11,12,24,13,39,29,55)(3,16,17,47,18,48,19,49)(4,21,22,25,23,54,30,58)(9,32,14,41,33,45,42,63)(10,34,35,38,37,56,36,53)(15,43,44,46,20,52,50,57)(31,59,51,61,60,64,40,62)Polycyclic presentation:
2^9|a^2=f,b^2=gh,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379154);