Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682648 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2170 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)(11,22)(12,27)(13,20)(14,19)(15,35)(16,21)(17,25)(18,31)(23,53)(24,52)(26,41)(28,39)(29,46)(30,59)(32,65)(33,47)(34,50)(36,42)(37,45)(38,49)(40,57)(43,54)(44,63)(48,58)(51,56)(55,72)(60,67)(61,71)(62,68)(64,70)(66,69);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,28)(13,32)(14,25)(15,26)(16,36)(18,43)(19,22)(20,48)(21,41)(23,31)(24,38)(27,51)(29,56)(30,62)(33,58)(34,57)(35,42)(37,54)(39,46)(40,49)(44,69)(45,53)(47,65)(50,52)(55,61)(59,72)(60,63)(64,66)(67,70)(68,71);(9,12,15,31,14,29,16,37)(10,18,21,27,19,45,35,46)(11,23,26,56,25,54,36,28)(13,30,34,63,33,61,38,64)(17,39,41,43,22,51,42,53)(20,44,49,59,47,70,50,71)(24,55,32,66,57,62,58,60)(40,67,65,68,52,69,48,72);(1,3,4,7)(2,6,5,8)(10,19)(12,29)(13,33)(17,22)(21,35)(23,54)(24,57)(28,56)(31,37)(32,58)(34,38)(41,42)(44,70)(59,71)(67,69)(68,72);(2,5)(6,8)(9,13,14,33)(10,20,19,47)(11,24,25,57)(12,30,29,61)(15,34,16,38)(17,40,22,52)(18,44,45,70)(21,49,35,50)(23,55,54,62)(26,32,36,58)(27,59,46,71)(28,60,56,66)(31,63,37,64)(39,67,51,69)(41,65,42,48)(43,68,53,72)Polycyclic presentation:
2^9|c^2=g,d^2=fi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590963);