Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682648 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1781 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)(9,10,16,21)(11,17,25,42)(12,26,28,46)(13,19,32,48)(14,34,15,20)(18,36,44,30)(22,43,51,64)(23,39,54,63)(24,41,35,40)(27,56,53,38)(29,57,59,68)(31,47,55,65)(33,50,37,49)(45,61,67,60)(52,66,69,72)(58,71,70,62);(1,2)(3,6)(4,8)(5,7)(9,11)(10,17)(12,27)(13,31)(14,24)(15,35)(16,25)(18,43)(19,47)(20,40)(21,42)(22,30)(23,33)(26,56)(28,53)(29,52)(32,55)(34,41)(36,51)(37,54)(38,46)(39,50)(44,64)(45,62)(48,65)(49,63)(57,66)(58,61)(59,69)(60,70)(67,71)(68,72);(9,12,14,30,16,28,15,36)(10,18,20,26,21,44,34,46)(11,22,24,53,25,51,35,27)(13,29,33,60,32,59,37,61)(17,38,40,43,42,56,41,64)(19,45,49,57,48,67,50,68)(23,52,55,70,54,69,31,58)(39,62,47,66,63,71,65,72);(1,3,5,4)(2,6,7,8)(12,28)(13,32)(18,44)(19,48)(22,51)(23,54)(26,46)(27,53)(30,36)(31,55)(33,37)(38,56)(39,63)(43,64)(47,65)(49,50);(1,4,5,3)(2,6,7,8)(9,13,16,32)(10,19,21,48)(11,23,25,54)(12,29,28,59)(14,33,15,37)(17,39,42,63)(18,45,44,67)(20,49,34,50)(22,52,51,69)(24,55,35,31)(26,57,46,68)(27,58,53,70)(30,60,36,61)(38,62,56,71)(40,47,41,65)(43,66,64,72)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=f,e^2=fi,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, 6594197);