Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6442587 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(9,10,15,20)(11,21,24,16)(12,26,30,54)(13,18,31,40)(14,33,27,19)(17,37,39,29)(22,45,48,52)(23,43,50,35)(25,36,46,44)(28,53,56,71)(32,58,55,41)(34,49,42,61)(38,62,63,57)(47,66,68,70)(51,60,67,65)(59,69,64,72);(1,2)(3,7)(4,5)(6,8)(9,11,14,25,15,24,27,46)(10,16,19,36,20,21,33,44)(12,22,29,49,30,48,37,61)(13,23,32,51,31,50,55,67)(17,34,26,52,39,42,54,45)(18,35,41,60,40,43,58,65)(28,47,57,69,56,68,62,72)(38,59,53,70,63,64,71,66);(2,5)(6,7)(9,12,27,37,15,30,14,29)(10,17,33,54,20,39,19,26)(11,22,46,61,24,48,25,49)(13,28,55,62,31,56,32,57)(16,34,44,45,21,42,36,52)(18,38,58,71,40,63,41,53)(23,47,67,72,50,68,51,69)(35,59,65,66,43,64,60,70);(1,3,4,8)(2,6,5,7)(9,13)(10,18)(11,23)(12,28)(14,32)(15,31)(16,35)(17,38)(19,41)(20,40)(21,43)(22,47)(24,50)(25,51)(26,53)(27,55)(29,57)(30,56)(33,58)(34,59)(36,60)(37,62)(39,63)(42,64)(44,65)(45,66)(46,67)(48,68)(49,69)(52,70)(54,71)(61,72);(13,31)(18,40)(23,50)(28,56)(32,55)(35,43)(38,63)(41,58)(47,68)(51,67)(53,71)(57,62)(59,64)(60,65)(66,70)(69,72)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gi,d^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370944);