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 | |
1632 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)(11,22)(12,27)(13,19)(14,17)(15,34)(16,23)(18,28)(20,33)(21,32)(24,36)(25,35)(26,38)(29,55)(30,44)(31,41)(37,53)(39,49)(40,58)(42,61)(43,66)(45,48)(46,54)(47,50)(51,65)(52,68)(56,60)(57,59)(62,69)(63,72)(64,70)(67,71);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,24)(13,23)(14,25)(15,26)(17,39)(18,36)(19,22)(20,37)(21,38)(27,45)(28,48)(29,51)(30,52)(31,49)(32,50)(33,54)(34,47)(35,41)(40,65)(42,46)(43,63)(44,64)(53,61)(55,62)(56,67)(57,68)(58,69)(59,70)(60,72)(66,71);(3,8)(5,7)(10,17)(11,23)(14,33)(15,32)(16,35)(18,40)(19,41)(20,21)(22,39)(24,48)(25,53)(27,55)(29,60)(30,59)(31,61)(34,42)(36,62)(37,47)(38,46)(43,44)(45,65)(49,54)(51,71)(56,57)(58,66)(63,68)(64,67)(69,72);(2,6)(5,7)(9,12)(10,18)(11,24)(13,28)(14,29)(15,30)(16,36)(17,40)(19,27)(20,43)(21,44)(22,45)(23,48)(25,51)(26,52)(31,58)(32,59)(33,60)(34,57)(35,62)(37,63)(38,64)(39,65)(41,55)(42,56)(46,67)(47,68)(49,69)(50,70)(53,71)(54,72)(61,66);(12,28)(18,27)(24,48)(29,58)(30,59)(36,45)(40,55)(43,56)(44,57)(51,69)(52,70)(60,66)(62,65)(63,67)(64,68)(71,72)Polycyclic presentation:
2^9|b^2=fi,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, 7530570);