Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623323 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1630 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)(3,7)(4,6)(5,8)(9,10)(11,23)(12,30)(13,20)(14,21)(15,37)(16,40)(17,27)(18,33)(19,32)(22,39)(24,53)(25,42)(26,54)(28,62)(29,43)(31,51)(34,50)(35,52)(36,46)(38,47)(41,59)(44,60)(45,57)(48,65)(49,66)(55,64)(56,72)(58,71)(61,67)(63,70)(68,69);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,25)(13,26)(14,27)(15,28)(16,29)(18,45)(19,42)(20,43)(21,23)(22,44)(24,32)(30,53)(31,58)(33,59)(34,56)(35,60)(36,61)(37,67)(38,68)(39,64)(40,54)(41,48)(46,62)(47,70)(49,69)(50,71)(51,72)(52,55)(57,65)(63,66);(3,8)(5,7)(9,12)(10,18)(11,24)(13,31)(14,32)(15,38)(16,34)(17,41)(19,46)(20,47)(21,48)(22,50)(23,45)(25,27)(26,56)(28,63)(29,58)(30,37)(33,39)(35,65)(36,66)(40,49)(42,67)(43,69)(44,72)(51,52)(53,62)(54,70)(55,71)(57,64)(59,60)(61,68);(2,6)(5,7)(9,12)(10,19)(11,25)(13,34)(14,32)(15,33)(16,31)(17,42)(18,46)(20,51)(21,30)(22,49)(23,53)(24,27)(26,56)(28,59)(29,58)(35,66)(36,65)(37,48)(38,39)(40,50)(41,67)(43,72)(44,69)(45,62)(47,52)(54,71)(55,70)(57,61)(60,63)(64,68);(9,13,14,16)(10,20,21,40)(11,26,27,29)(12,31,32,34)(15,35,36,39)(17,43,23,54)(18,47,48,49)(19,50,30,51)(22,37,52,46)(24,56,25,58)(28,60,61,64)(33,38,65,66)(41,69,45,70)(42,71,53,72)(44,67,55,62)(57,63,59,68)Polycyclic presentation:
2^9|b^2=fi,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gh,h^a=hi,c^b=cf,g^c=gh,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530574);