Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234321 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2619 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1201 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18,21,24)(19,22,26,30)(20,23,28,32)(25,29,27,31)(33,34)(35,44)(36,47)(37,42)(38,43)(39,40)(41,50)(45,56)(46,57)(48,60)(49,53)(51,54)(52,58)(55,61)(59,64)(62,63);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19,21,26)(18,22,24,30)(20,27,28,25)(23,31,32,29)(33,35,38,39)(34,40,43,44)(36,45,49,50)(37,51,52,46)(41,53,56,47)(42,57,58,54)(48,61,62,59)(55,60,64,63);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,20)(18,23)(19,25)(21,28)(22,29)(24,32)(26,27)(30,31)(33,36,35,45,38,49,39,50)(34,41,40,53,43,56,44,47)(37,48,46,59,52,62,51,61)(42,55,54,63,58,64,57,60);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,16,10,15)(20,28)(23,32)(25,27)(29,31)(33,37)(34,42)(35,46)(36,48)(38,52)(39,51)(40,54)(41,55)(43,58)(44,57)(45,59)(47,60)(49,62)(50,61)(53,63)(56,64);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(19,26)(22,30)(25,27)(29,31)(34,43)(37,52)(40,44)(41,56)(46,51)(47,53)(48,62)(59,61)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=fg,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7179275);