Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437602 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2053 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,17)(12,26)(13,19)(14,35)(15,21)(16,20)(18,29)(22,51)(23,39)(24,42)(25,40)(27,38)(28,57)(30,45)(31,50)(32,46)(33,49)(34,47)(36,41)(37,48)(43,53)(44,59)(52,69)(54,65)(55,64)(56,63)(58,62)(60,68)(61,67)(66,71)(70,72);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,27)(13,32)(14,24)(15,36)(16,25)(18,43)(19,46)(20,40)(21,41)(22,30)(23,33)(26,38)(28,52)(29,53)(31,56)(34,54)(35,42)(37,55)(39,49)(44,62)(45,51)(47,65)(48,64)(50,63)(57,69)(58,59)(60,71)(61,70)(66,68)(67,72);(9,12,16,31)(10,18,20,45)(11,22,25,53)(13,28,34,60)(14,30,15,29)(17,38,40,63)(19,44,47,67)(21,50,35,26)(23,52,55,71)(24,56,36,27)(32,58,54,70)(33,61,37,59)(39,62,64,72)(41,43,42,51)(46,66,65,69)(48,68,49,57);(1,4,5,3)(2,6,7,8)(9,13)(10,19)(11,23)(12,28)(14,33)(15,37)(16,34)(17,39)(18,44)(20,47)(21,48)(22,52)(24,54)(25,55)(26,57)(27,58)(29,59)(30,61)(31,60)(32,36)(35,49)(38,62)(40,64)(41,46)(42,65)(43,66)(45,67)(50,68)(51,69)(53,71)(56,70)(63,72);(10,20)(17,40)(18,45)(19,47)(21,35)(26,50)(38,63)(39,64)(41,42)(43,51)(44,67)(46,65)(48,49)(57,68)(62,72)(66,69)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346931);