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 | #532 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1244 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2,5,4)(3,7,8,6)(9,10,15,19)(11,16,22,37)(12,18,26,44)(13,17,31,40)(14,32,30,21)(20,45,41,29)(23,34,46,59)(24,38,47,33)(25,53,48,36)(27,39,54,65)(28,57,55,43)(35,51,62,52)(42,66,64,56)(49,63,67,58)(50,71,68,61)(60,69,72,70);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,23)(13,24)(14,25)(15,22)(17,38)(18,34)(19,37)(20,35)(21,36)(26,46)(27,49)(28,50)(29,52)(30,48)(31,47)(32,53)(33,40)(39,63)(41,62)(42,60)(43,61)(44,59)(45,51)(54,67)(55,68)(56,70)(57,71)(58,65)(64,72)(66,69);(1,4,5,2)(3,7,8,6)(10,17)(11,22)(13,29)(14,30)(16,33)(18,39)(19,40)(20,21)(23,46)(24,51)(27,56)(28,55)(31,45)(32,41)(34,58)(35,53)(36,62)(37,38)(42,43)(44,65)(47,52)(49,69)(54,66)(57,64)(59,63)(60,71)(61,72)(67,70);(3,8)(6,7)(9,12)(10,18)(11,23)(13,27)(14,28)(15,26)(16,34)(17,39)(19,44)(20,42)(21,43)(22,46)(24,49)(25,50)(29,56)(30,55)(31,54)(32,57)(33,58)(35,60)(36,61)(37,59)(38,63)(40,65)(41,64)(45,66)(47,67)(48,68)(51,69)(52,70)(53,71)(62,72);(10,19)(11,22)(12,26)(17,40)(20,41)(21,32)(24,47)(25,48)(27,54)(28,55)(34,59)(51,52)(56,66)(58,63)(60,72)(61,71)Polycyclic presentation:
2^9|a^2=fi,c^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,c^b=ci,d^b=df,e^b=ei,g^c=gh,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531047);