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 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(2,6)(7,8)(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)(28,61)(29,62)(31,40)(32,46)(33,60)(35,43)(38,47)(39,67)(41,64)(44,63)(45,65)(48,53)(49,56)(51,59)(52,71)(54,69)(55,66)(58,68)(70,72);(1,2,5,6)(3,7,4,8)(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)(27,59,62,50)(28,54,38,52)(30,53,63,67)(32,64,36,56)(34,66,60,57)(35,58,65,72)(43,70,45,68)(47,69,61,71);(1,3,5,4)(2,7,6,8)(9,12,14,29)(10,18,20,44)(11,23,24,51)(13,28,16,38)(15,35,33,65)(17,39,22,48)(19,43,37,45)(21,47,46,61)(25,52,31,54)(26,58,55,72)(27,60,62,34)(30,32,63,36)(40,68,42,70)(41,69,49,71)(50,66,59,57)(53,64,67,56);(1,4,5,3)(2,7,6,8)(11,24)(17,22)(23,51)(25,31)(26,55)(39,48)(40,42)(41,49)(50,59)(52,54)(53,67)(56,64)(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)(21,34,46,60)(23,52,51,54)(26,56,55,64)(27,61,62,47)(30,35,63,65)(39,68,48,70)(41,57,49,66)(50,71,59,69)(53,72,67,58)Polycyclic presentation:
2^9|b^2=f,c^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, 7531167);