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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1764 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)(3,12)(4,9)(5,10)(6,20)(7,11)(8,15)(13,26)(14,22)(16,27)(17,23)(18,24)(19,25)(21,29)(28,32)(30,31)(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);(1,3,5,14,7,16,6,15)(2,8,20,27,11,22,10,12)(4,13,19,30,17,28,18,29)(9,21,24,32,23,31,25,26)(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);(4,17)(9,23)(13,28)(18,19)(21,31)(24,25)(26,32)(29,30)(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)(2,9)(3,13)(5,19)(6,18)(7,17)(8,21)(10,25)(11,23)(12,26)(14,30)(15,29)(16,28)(20,24)(22,31)(27,32)(36,47)(40,55)(43,57)(46,61)(48,54)(51,63)(56,64)(58,62);(1,5,7,6)(2,10,11,20)(3,14,16,15)(4,18,17,19)(8,12,22,27)(9,24,23,25)(13,29,28,30)(21,32,31,26)(35,44)(39,52)(43,57)(45,59)(51,63)(53,60)(56,64)(58,62)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=g,e^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, 6265251);