Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6473167 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1866 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)(3,8)(4,6)(5,7)(9,10,16,22)(11,23,27,46)(12,28,32,41)(13,19,33,43)(14,21,15,20)(17,36,39,26)(18,30,42,31)(24,49,52,37)(25,48,53,66)(29,55,56,64)(34,45,35,44)(38,60,62,54)(40,57,63,58)(47,59,65,51)(50,68,69,61)(67,71,72,70);(3,4)(6,8)(9,11)(10,17)(12,24)(13,25)(14,36)(15,26)(16,27)(18,37)(19,38)(20,46)(21,23)(22,39)(28,47)(29,50)(30,59)(31,51)(32,52)(33,53)(34,60)(35,54)(40,61)(41,65)(42,49)(43,62)(44,66)(45,48)(55,67)(56,69)(57,71)(58,70)(63,68)(64,72);(1,3,5,4)(2,6,7,8)(9,12,14,30,16,32,15,31)(10,18,20,41,22,42,21,28)(11,24,36,59,27,52,26,51)(13,29,34,57,33,56,35,58)(17,37,46,65,39,49,23,47)(19,40,44,64,43,63,45,55)(25,50,60,71,53,69,54,70)(38,61,66,72,62,68,48,67);(1,4,5,3)(2,6,7,8)(9,13,16,33)(10,19,22,43)(11,25,27,53)(12,29,32,56)(14,34,15,35)(17,38,39,62)(18,40,42,63)(20,44,21,45)(23,48,46,66)(24,50,52,69)(26,54,36,60)(28,55,41,64)(30,57,31,58)(37,61,49,68)(47,67,65,72)(51,70,59,71);(2,7)(6,8)(13,33)(19,43)(25,53)(29,56)(34,35)(38,62)(40,63)(44,45)(48,66)(50,69)(54,60)(55,64)(57,58)(61,68)(67,72)(70,71)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6390492);