Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624842 |
| 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 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1269 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)(3,7)(4,6)(5,8)(9,10,15,20)(11,16,22,36)(12,25,30,56)(13,31,33,63)(14,34,32,19)(17,39,42,28)(18,43,44,27)(21,45,26,53)(23,51,49,67)(24,52,50,38)(29,62,61,55)(35,58,40,47)(37,57,65,46)(41,60,54,66)(48,71,59,69)(64,70,68,72);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,26)(13,23)(14,24)(15,22)(17,40)(18,37)(19,38)(20,36)(21,30)(25,53)(27,46)(28,58)(29,59)(31,51)(32,50)(33,49)(34,52)(35,42)(39,47)(41,68)(43,57)(44,65)(45,56)(48,61)(54,64)(55,71)(60,72)(62,69)(63,67)(66,70);(9,12,27,60,14,29,33,39,15,30,43,66,32,61,13,28)(10,17,31,55,19,41,44,56,20,42,63,62,34,54,18,25)(11,21,46,70,24,48,49,58,22,26,57,72,50,59,23,47)(16,35,51,69,38,64,65,53,36,40,67,71,52,68,37,45);(3,8)(5,7)(9,11)(10,16)(12,21)(13,23)(14,24)(15,22)(17,35)(18,37)(19,38)(20,36)(25,45)(26,30)(27,46)(28,47)(29,48)(31,51)(32,50)(33,49)(34,52)(39,58)(40,42)(41,64)(43,57)(44,65)(53,56)(54,68)(55,69)(59,61)(60,70)(62,71)(63,67)(66,72);(2,6)(5,7)(11,22)(16,36)(21,26)(23,49)(24,50)(35,40)(37,65)(38,52)(45,53)(46,57)(47,58)(48,59)(51,67)(64,68)(69,71)(70,72)Polycyclic presentation:
2^9|a^2=i,c^2=gh,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,c^b=ci,d^b=df,e^b=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532061);