Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499879 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23900 |
| 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 | |
1815 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)(3,15)(4,10)(5,13)(6,25)(7,30)(8,14)(9,18)(11,24)(12,29)(16,48)(17,34)(19,43)(20,55)(21,45)(22,58)(23,41)(26,36)(27,39)(28,42)(31,47)(32,40)(33,53)(35,54)(37,56)(38,57)(44,52)(46,51)(49,64)(50,61)(59,62)(60,63);(1,3,8,19)(2,9,14,36)(4,20,23,49)(5,17,7,31)(6,26,29,18)(10,37,41,62)(11,34,42,47)(12,43,25,15)(13,44,30,35)(16,32,51,21)(22,59,27,56)(24,54,28,52)(33,45,61,40)(38,48,63,46)(39,64,58,55)(50,57,53,60);(1,4)(2,10)(3,16)(5,21)(6,27)(7,32)(8,23)(9,33)(11,38)(12,39)(13,45)(14,41)(15,46)(17,49)(18,53)(19,51)(20,31)(22,29)(24,60)(25,58)(26,50)(28,57)(30,40)(34,55)(35,62)(36,61)(37,44)(42,63)(43,48)(47,64)(52,56)(54,59);(1,5,8,7)(2,11,14,42)(3,17,19,31)(4,21,23,32)(6,28,29,24)(9,34,36,47)(10,38,41,63)(12,30,25,13)(15,44,43,35)(16,49,51,20)(18,54,26,52)(22,60,27,57)(33,55,61,64)(37,48,62,46)(39,40,58,45)(50,56,53,59)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fh,g^b=gi,e^c=eh,g^c=gi,e^d=ei,f^d=fhi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72312286, #72312329, #72312376, … |
| maximal normal subgroup of | #72312286, #72312329, #72312376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407080);