Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6358021 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #61006 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,8)(3,11,15,24)(4,9,16,22)(5,18,12,10)(7,19,20,14)(13,25,27,32)(17,29,26,23)(21,30,31,28)(33,34,38,42)(35,39,44,52)(36,46,47,61)(37,49,50,41)(40,54,55,48)(43,56,57,64)(45,60,59,53)(51,62,63,58);(1,3,12,19,6,15,5,14)(2,7,18,24,8,20,10,11)(4,13,26,30,16,27,17,28)(9,21,29,32,22,31,23,25)(33,35)(34,39)(36,43)(37,45)(38,44)(40,51)(41,53)(42,52)(46,56)(47,57)(48,58)(49,60)(50,59)(54,62)(55,63)(61,64);(2,8)(4,16)(7,20)(10,18)(11,24)(13,27)(17,26)(28,30)(33,36,37,48,38,47,50,54)(34,40,41,46,42,55,49,61)(35,43,45,58,44,57,59,62)(39,51,53,56,52,63,60,64);(1,4,6,16)(2,9,8,22)(3,13,15,27)(5,17,12,26)(7,21,20,31)(10,23,18,29)(11,25,24,32)(14,28,19,30)(36,47)(40,55)(43,57)(46,61)(48,54)(51,63)(56,64)(58,62);(4,16)(9,22)(13,27)(17,26)(21,31)(23,29)(25,32)(28,30)(35,44)(39,52)(43,57)(45,59)(51,63)(53,60)(56,64)(58,62)Polycyclic presentation:
2^9|a^2=hi,b^2=fh,c^2=gh,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,e^b=ei,d^c=dhi,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265248);