Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021375 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #61081 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(3,7)(6,8)(9,10,16,21,15,22,14,20)(11,23,27,17,28,48,26,40)(12,29,36,43,32,42,31,18)(13,19,37,47,34,45,35,46)(24,50,53,49,54,38,52,67)(25,44,56,39,55,68,33,61)(30,57,62,65,59,66,60,41)(51,69,71,64,70,63,58,72);(2,5)(3,7)(9,11)(10,17)(12,24)(13,33)(14,27)(15,28)(16,26)(18,38)(19,44)(20,48)(21,23)(22,40)(25,35)(29,49)(30,58)(31,53)(32,54)(34,56)(36,52)(37,55)(39,46)(41,64)(42,67)(43,50)(45,68)(47,61)(51,60)(57,69)(59,71)(62,70)(63,66)(65,72);(1,2)(3,6)(4,5)(7,8)(9,12,16,36,15,32,14,31)(10,18,20,42,22,43,21,29)(11,24,26,52,28,54,27,53)(13,30,37,62,34,59,35,60)(17,38,48,67,40,50,23,49)(19,41,46,66,45,65,47,57)(25,51,33,58,55,70,56,71)(39,63,68,72,61,69,44,64);(2,5)(6,8)(9,13,16,37,15,34,14,35)(10,19,21,47,22,45,20,46)(11,25,27,56,28,55,26,33)(12,30,36,62,32,59,31,60)(17,39,48,68,40,61,23,44)(18,41,29,57,43,65,42,66)(24,51,53,71,54,70,52,58)(38,63,67,72,50,69,49,64);(1,3,4,7)(2,6,5,8)(13,34)(19,45)(25,55)(30,59)(33,56)(35,37)(39,61)(41,65)(44,68)(46,47)(51,70)(57,66)(58,71)(60,62)(63,69)(64,72)Polycyclic presentation:
2^9|a^2=fi,c^2=gi,d^2=fi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7063414);