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 | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,22)(12,27)(13,19)(14,20)(15,36)(16,21)(17,25)(18,31)(23,43)(24,52)(26,41)(28,51)(29,58)(30,50)(32,45)(33,63)(34,47)(35,49)(37,42)(38,48)(39,57)(40,55)(44,61)(46,56)(53,66)(54,65)(59,69)(60,68)(62,67)(64,71)(70,72);(2,5)(6,8)(9,11)(10,17)(12,28)(13,33)(14,25)(15,26)(16,37)(18,43)(19,46)(20,22)(21,41)(23,32)(24,35)(27,39)(29,53)(30,57)(31,54)(34,56)(36,42)(38,55)(40,49)(44,64)(45,65)(47,63)(48,52)(50,51)(58,72)(59,61)(60,70)(62,71)(66,68)(67,69);(9,12,14,30)(10,18,20,45)(11,23,25,54)(13,29,34,60)(15,32,16,31)(17,39,22,51)(19,44,47,67)(21,50,36,27)(24,53,55,70)(26,57,37,28)(33,59,56,71)(35,62,38,61)(40,64,52,69)(41,43,42,65)(46,66,63,72)(48,68,49,58);(1,3,4,7)(2,6,5,8)(9,13,14,34)(10,19,20,47)(11,24,25,55)(12,29,30,60)(15,35,16,38)(17,40,22,52)(18,44,45,67)(21,48,36,49)(23,53,54,70)(26,56,37,33)(27,58,50,68)(28,59,57,71)(31,61,32,62)(39,64,51,69)(41,46,42,63)(43,66,65,72);(11,25)(17,22)(23,54)(24,55)(26,37)(28,57)(33,56)(39,51)(40,52)(41,42)(43,65)(46,63)(53,70)(59,71)(64,69)(66,72)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,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, 6346811);