Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6358057 |
| 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 | #3511 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,10)(3,11,12,24)(4,8,15,22)(5,17,18,9)(7,19,20,14)(13,25,26,32)(16,29,28,23)(21,30,31,27)(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);(1,3,5,14,6,12,18,19)(2,7,9,11,10,20,17,24)(4,13,16,27,15,26,28,30)(8,21,23,25,22,31,29,32)(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);(3,12)(4,15)(7,20)(8,22)(11,24)(14,19)(16,28)(23,29)(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)(2,8)(3,13)(5,16)(6,15)(7,21)(9,23)(10,22)(11,25)(12,26)(14,27)(17,29)(18,28)(19,30)(20,31)(24,32)(36,47)(40,55)(43,57)(46,61)(48,54)(51,63)(56,64)(58,62);(4,15)(8,22)(13,26)(16,28)(21,31)(23,29)(25,32)(27,30)(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=f,c^2=g,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,e^b=ei,d^c=dhi,e^d=eh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265274);