Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6473174 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2042 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(3,7)(6,8)(9,10,16,22)(11,23,27,46)(12,28,30,42)(13,19,33,43)(14,21,15,20)(17,36,39,26)(18,31,41,32)(24,49,51,37)(25,48,53,66)(29,55,58,63)(34,45,35,44)(38,60,62,54)(40,56,64,57)(47,59,65,52)(50,68,70,61)(67,71,72,69);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,24)(13,25)(14,36)(15,26)(16,27)(18,37)(19,38)(20,46)(21,23)(22,39)(28,47)(29,50)(30,51)(31,59)(32,52)(33,53)(34,60)(35,54)(40,61)(41,49)(42,65)(43,62)(44,66)(45,48)(55,67)(56,71)(57,69)(58,70)(63,72)(64,68);(2,5)(6,8)(9,12,14,31,16,30,15,32)(10,18,20,42,22,41,21,28)(11,24,36,59,27,51,26,52)(13,29,34,56,33,58,35,57)(17,37,46,65,39,49,23,47)(19,40,44,63,43,64,45,55)(25,50,60,71,53,70,54,69)(38,61,66,72,62,68,48,67);(2,5)(6,8)(9,13)(10,19)(11,25)(12,29)(14,34)(15,35)(16,33)(17,38)(18,40)(20,44)(21,45)(22,43)(23,48)(24,50)(26,54)(27,53)(28,55)(30,58)(31,56)(32,57)(36,60)(37,61)(39,62)(41,64)(42,63)(46,66)(47,67)(49,68)(51,70)(52,69)(59,71)(65,72);(1,3,4,7)(2,6,5,8)(12,30)(13,33)(18,41)(19,43)(24,51)(25,53)(28,42)(31,32)(34,35)(37,49)(38,62)(44,45)(47,65)(48,66)(52,59)(54,60)Polycyclic presentation:
2^9|a^2=i,c^2=gi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6401269);