Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623311 |
| 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 | #33612 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1295 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(2,5)(3,7)(9,10)(11,21)(12,26)(13,18)(14,33)(15,36)(16,23)(17,29)(19,35)(20,32)(22,49)(24,54)(25,40)(27,60)(28,59)(30,64)(31,58)(34,46)(37,45)(38,61)(39,69)(41,50)(42,62)(43,65)(44,63)(47,56)(48,53)(51,70)(52,66)(55,68)(57,72)(67,71);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,27)(13,23)(14,24)(15,25)(17,41)(18,21)(19,39)(20,40)(22,28)(26,49)(29,61)(30,57)(31,52)(32,53)(33,66)(34,67)(35,56)(36,48)(37,51)(38,42)(43,71)(44,47)(45,72)(46,68)(50,62)(54,58)(55,65)(59,60)(63,69)(64,70);(2,5)(6,8)(9,12,13,28)(10,17,18,42)(11,22,23,27)(14,34,31,65)(15,37,32,30)(16,38,21,41)(19,45,44,64)(20,46,36,43)(24,55,52,67)(25,57,53,51)(26,58,59,33)(29,63,62,35)(39,70,47,72)(40,71,48,68)(49,66,60,54)(50,69,61,56);(1,3,4,7)(2,6,5,8)(10,18)(16,21)(17,42)(19,44)(20,36)(26,59)(33,58)(38,41)(39,47)(40,48)(43,46)(45,64)(49,60)(54,66)(68,71)(70,72);(11,23)(16,21)(22,27)(24,52)(25,53)(38,41)(39,47)(40,48)(49,60)(50,61)(51,57)(54,66)(55,67)(56,69)(68,71)(70,72)Polycyclic presentation:
2^9|c^2=i,d^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gh,h^a=hi,c^b=cf,e^b=ei,g^c=gh,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530678);