Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624184 |
| 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 | #61051 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1498 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)(3,7)(4,6)(5,8)(9,10,16,22)(11,17,25,40)(12,27,30,59)(13,21,15,19)(14,33,36,46)(18,42,44,29)(20,32,48,35)(23,50,52,71)(24,45,31,39)(26,57,56,69)(28,60,37,47)(34,49,62,43)(38,63,65,53)(41,61,67,55)(51,68,54,70)(58,64,72,66);(1,3)(2,5)(4,8)(6,7)(9,11,16,25)(10,17,22,40)(12,23,30,52)(13,31,15,24)(14,26,36,56)(18,38,44,65)(19,45,21,39)(20,41,48,67)(27,50,59,71)(28,54,37,51)(29,53,42,63)(32,61,35,55)(33,57,46,69)(34,58,62,72)(43,66,49,64)(47,68,60,70);(9,12)(10,18)(11,23)(13,28)(14,34)(15,37)(16,30)(17,38)(19,43)(20,47)(21,49)(22,44)(24,51)(25,52)(26,58)(27,33)(29,35)(31,54)(32,42)(36,62)(39,64)(40,65)(41,68)(45,66)(46,59)(48,60)(50,57)(53,55)(56,72)(61,63)(67,70)(69,71);(3,8)(5,7)(9,13,16,15)(10,19,22,21)(11,24,25,31)(12,28,30,37)(14,32,36,35)(17,39,40,45)(18,43,44,49)(20,46,48,33)(23,51,52,54)(26,55,56,61)(27,47,59,60)(29,34,42,62)(38,64,65,66)(41,57,67,69)(50,70,71,68)(53,72,63,58);(2,6)(5,7)(11,25)(17,40)(23,52)(24,31)(26,56)(38,65)(39,45)(41,67)(50,71)(51,54)(53,63)(55,61)(57,69)(58,72)(64,66)(68,70)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,e^b=ei,g^c=gh,h^c=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531386);