Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623613 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | #525 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,17)(12,26)(13,20)(14,32)(15,36)(16,19)(18,28)(21,34)(22,48)(23,40)(24,54)(25,39)(27,58)(29,57)(30,42)(31,46)(33,44)(35,61)(37,47)(38,50)(41,53)(43,59)(45,62)(49,67)(51,72)(52,69)(55,68)(56,66)(60,65)(63,71)(64,70);(1,3)(2,6)(4,7)(5,8)(9,11,16,25)(10,17,19,39)(12,22,29,52)(13,30,15,23)(14,24,35,56)(18,38,43,63)(20,42,36,40)(21,41,46,65)(26,48,57,69)(27,51,37,49)(28,50,59,71)(31,60,34,53)(32,54,61,66)(33,55,62,70)(44,68,45,64)(47,67,58,72);(1,4,5,2)(3,7,8,6)(9,12)(10,18)(11,22)(13,27)(14,33)(15,37)(16,29)(17,38)(19,43)(20,44)(21,47)(23,49)(24,55)(25,52)(26,32)(28,34)(30,51)(31,59)(35,62)(36,45)(39,63)(40,64)(41,67)(42,68)(46,58)(48,54)(50,53)(56,70)(57,61)(60,71)(65,72)(66,69);(3,8)(6,7)(10,19)(17,39)(18,43)(20,36)(21,46)(26,57)(32,61)(38,63)(40,42)(41,65)(44,45)(47,58)(48,69)(54,66)(64,68)(67,72);(9,13,16,15)(10,20,19,36)(11,23,25,30)(12,27,29,37)(14,31,35,34)(17,40,39,42)(18,44,43,45)(21,32,46,61)(22,49,52,51)(24,53,56,60)(26,58,57,47)(28,33,59,62)(38,64,63,68)(41,66,65,54)(48,67,69,72)(50,70,71,55)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531024);