Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6814827 |
| 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 |
| prime radical | #61088 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2002 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)(9,10,15,21)(11,16,26,40)(12,27,29,45)(13,18,36,50)(14,20,34,19)(17,41,43,31)(22,42,52,69)(23,38,57,68)(24,51,25,39)(28,58,54,37)(30,59,33,63)(32,46,56,72)(35,49,65,48)(44,64,47,61)(53,70,55,71)(60,67,62,66);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,28)(13,32)(14,24)(15,26)(17,42)(18,46)(19,39)(20,51)(21,40)(22,31)(23,35)(25,34)(27,58)(29,54)(30,55)(33,53)(36,56)(37,45)(38,49)(41,52)(43,69)(44,67)(47,66)(48,68)(50,72)(57,65)(59,71)(60,61)(62,64)(63,70);(9,12,15,29)(10,17)(11,22)(13,33,36,30)(14,31,34,41)(16,37,40,58)(18,47)(19,27)(20,45)(21,43)(23,55)(24,54)(25,28)(26,52)(32,62)(35,64,65,61)(38,67,68,66)(39,42,51,69)(44,50)(46,71,72,70)(48,63)(49,59)(53,57)(56,60);(1,3,5,4)(2,6,7,8)(12,29)(17,43)(22,52)(27,45)(28,54)(30,33)(31,41)(37,58)(42,69)(44,47)(53,55)(59,63)(60,62)(61,64)(66,67)(70,71);(1,4,5,3)(2,6,7,8)(9,13,34,65,15,36,14,35)(10,18,19,48,21,50,20,49)(11,23,24,56,26,57,25,32)(12,30,41,64,29,33,31,61)(16,38,51,72,40,68,39,46)(17,44,27,59,43,47,45,63)(22,53,54,62,52,55,28,60)(37,66,69,71,58,67,42,70)Polycyclic presentation:
2^9|a^2=fi,c^2=gh,d^2=f,e^2=fhi,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,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724163);