Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433077 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2696 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)(3,5)(4,6)(7,8)(9,10,16,22)(11,23,28,39)(12,29,32,46)(13,19,33,43)(14,21,15,20)(17,27,40,26)(18,36,42,31)(24,49,52,37)(25,48,53,63)(30,57,58,66)(34,45,35,44)(38,55,62,54)(41,60,65,59)(47,56,64,51)(50,68,69,61)(67,71,72,70);(1,3)(2,5)(4,7)(6,8)(9,11,15,27,16,28,14,26)(10,17,21,39,22,40,20,23)(12,24,36,56,32,52,31,51)(13,25,35,55,33,53,34,54)(18,37,46,64,42,49,29,47)(19,38,45,63,43,62,44,48)(30,50,60,71,58,69,59,70)(41,61,66,72,65,68,57,67);(3,7)(5,8)(9,12)(10,18)(11,24)(13,30)(14,31)(15,36)(16,32)(17,37)(19,41)(20,29)(21,46)(22,42)(23,47)(25,50)(26,51)(27,56)(28,52)(33,58)(34,59)(35,60)(38,61)(39,64)(40,49)(43,65)(44,57)(45,66)(48,67)(53,69)(54,70)(55,71)(62,68)(63,72);(2,6)(5,8)(9,13)(10,19)(11,25)(12,30)(14,34)(15,35)(16,33)(17,38)(18,41)(20,44)(21,45)(22,43)(23,48)(24,50)(26,54)(27,55)(28,53)(29,57)(31,59)(32,58)(36,60)(37,61)(39,63)(40,62)(42,65)(46,66)(47,67)(49,68)(51,70)(52,69)(56,71)(64,72);(13,33)(19,43)(25,53)(30,58)(34,35)(38,62)(41,65)(44,45)(48,63)(50,69)(54,55)(57,66)(59,60)(61,68)(67,72)(70,71)Polycyclic presentation:
2^9|a^2=i,b^2=fi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340275);