Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623311 |
| 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 | |
1623 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)(9,10,13,19)(11,22,23,16)(12,27,28,18)(14,17,31,41)(15,34,32,21)(20,45,42,33)(24,36,49,46)(25,35,50,39)(26,38,51,48)(29,56,59,40)(30,44,60,58)(37,55,47,54)(43,61,57,66)(52,65,69,62)(53,68,70,64)(63,71,67,72);(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)(27,46)(28,49)(29,52)(30,53)(31,50)(32,51)(33,55)(34,48)(35,41)(40,65)(42,47)(43,63)(44,64)(45,54)(56,62)(57,67)(58,68)(59,69)(60,70)(61,72)(66,71);(3,8)(5,7)(10,17)(11,23)(14,33)(15,32)(16,35)(18,40)(19,41)(20,21)(22,39)(24,49)(25,54)(27,56)(29,61)(30,60)(31,45)(34,42)(36,62)(37,48)(38,47)(43,44)(46,65)(50,55)(52,71)(57,58)(59,66)(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)(22,46)(23,49)(25,52)(26,53)(31,59)(32,60)(33,61)(34,58)(35,62)(37,63)(38,64)(39,65)(41,56)(42,57)(45,66)(47,67)(48,68)(50,69)(51,70)(54,71)(55,72);(11,23)(12,28)(16,22)(18,27)(25,50)(26,51)(29,59)(30,60)(35,39)(37,47)(38,48)(40,56)(43,57)(44,58)(54,55)(61,66)Polycyclic presentation:
2^9|a^2=i,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,e^b=ei,g^c=gh,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530586);