Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501150 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24404 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1057 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,15,7,14)(3,16,19,47,20,44,32,62)(4,21,25,41,23,40,26,10)(5,12,30,45,29,11,31,43)(9,33,35,18,37,52,36,27)(17,48,22,56,49,64,53,46)(24,55,59,39,28,61,60,42)(34,50,63,58,54,51,38,57);(1,3)(2,9)(4,22)(5,27)(6,19)(7,32)(8,20)(10,38)(11,16)(12,44)(13,36)(14,35)(15,37)(17,26)(18,29)(21,54)(23,53)(24,58)(25,49)(28,57)(30,33)(31,52)(34,40)(39,64)(41,63)(42,48)(43,62)(45,47)(46,55)(50,60)(51,59)(56,61);(1,4)(2,10)(3,17)(5,28)(6,26)(7,25)(8,23)(9,34)(11,42)(12,39)(13,40)(14,21)(15,41)(16,46)(18,51)(19,53)(20,49)(22,32)(24,29)(27,50)(30,59)(31,60)(33,57)(35,38)(36,63)(37,54)(43,61)(44,56)(45,55)(47,64)(48,62)(52,58);(2,11)(4,23)(5,29)(9,16)(10,39)(12,15)(13,45)(14,43)(17,49)(18,27)(21,55)(22,53)(25,26)(30,31)(33,52)(34,56)(35,62)(36,47)(37,44)(38,64)(40,61)(41,42)(46,54)(48,63)Polycyclic presentation:
2^9|a^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,d^c=di,e^c=eh,f^c=fi,g^c=gi,e^d=ei,f^d=fhi,f^e=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408346);