Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500489 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
1872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(9,10,16,22)(11,23,28,39)(12,29,32,46)(13,19,33,43)(14,21,15,20)(17,27,40,26)(18,36,42,31)(24,49,52,37)(25,48,53,63)(30,57,58,66)(34,45,35,44)(38,55,62,54)(41,60,65,59)(47,56,64,51)(50,68,69,61)(67,71,72,70);(1,2)(3,6)(4,5)(7,8)(9,11,15,27,16,28,14,26)(10,17,21,39,22,40,20,23)(12,24,36,56,32,52,31,51)(13,25,35,55,33,53,34,54)(18,37,46,64,42,49,29,47)(19,38,45,63,43,62,44,48)(30,50,60,71,58,69,59,70)(41,61,66,72,65,68,57,67);(2,5)(3,7)(9,12)(10,18)(11,24)(13,30)(14,31)(15,36)(16,32)(17,37)(19,41)(20,29)(21,46)(22,42)(23,47)(25,50)(26,51)(27,56)(28,52)(33,58)(34,59)(35,60)(38,61)(39,64)(40,49)(43,65)(44,57)(45,66)(48,67)(53,69)(54,70)(55,71)(62,68)(63,72);(1,3,4,7)(2,6,5,8)(9,13,16,33)(10,19,22,43)(11,25,28,53)(12,30,32,58)(14,34,15,35)(17,38,40,62)(18,41,42,65)(20,44,21,45)(23,48,39,63)(24,50,52,69)(26,54,27,55)(29,57,46,66)(31,59,36,60)(37,61,49,68)(47,67,64,72)(51,70,56,71);(2,5)(6,8)(13,33)(19,43)(25,53)(30,58)(34,35)(38,62)(41,65)(44,45)(48,63)(50,69)(54,55)(57,66)(59,60)(61,68)(67,72)(70,71)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6420799);