Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623604 |
| 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 | C2×#508 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1212 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)(11,22)(12,27)(13,19)(14,20)(15,34)(16,37)(17,24)(18,30)(21,36)(23,50)(25,42)(26,57)(28,60)(29,61)(31,40)(32,46)(33,64)(35,43)(38,47)(39,67)(41,63)(44,62)(45,65)(48,53)(49,56)(51,59)(52,71)(54,69)(55,66)(58,68)(70,72);(1,3,5,8)(2,6,4,7)(9,11,14,24)(10,17,20,22)(12,23,29,51)(13,31,16,25)(15,26,33,55)(18,39,44,48)(19,42,37,40)(21,41,46,49)(27,59,61,50)(28,54,38,52)(30,53,62,67)(32,63,36,56)(34,66,64,57)(35,58,65,72)(43,70,45,68)(47,69,60,71);(9,12)(10,18)(11,23)(13,28)(14,29)(15,35)(16,38)(17,39)(19,43)(20,44)(21,47)(22,48)(24,51)(25,52)(26,58)(27,34)(30,36)(31,54)(32,62)(33,65)(37,45)(40,68)(41,69)(42,70)(46,60)(49,71)(50,57)(53,56)(55,72)(59,66)(61,64)(63,67);(1,4,5,2)(3,6,8,7)(11,24)(17,22)(23,51)(25,31)(26,55)(39,48)(40,42)(41,49)(50,59)(52,54)(53,67)(56,63)(57,66)(58,72)(68,70)(69,71);(9,13,14,16)(10,19,20,37)(11,25,24,31)(12,28,29,38)(15,32,33,36)(17,40,22,42)(18,43,44,45)(21,34,46,64)(23,52,51,54)(26,56,55,63)(27,60,61,47)(30,35,62,65)(39,68,48,70)(41,57,49,66)(50,71,59,69)(53,72,67,58)Polycyclic presentation:
2^9|a^2=fi,b^2=f,d^2=fi,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531131);