Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #497333 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #61090 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#576 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1121 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)(3,13)(4,19,22,36)(5,23)(6,26,28,11)(7,12)(8,16)(9,21)(10,24)(14,39,43,63)(15,32)(17,46,48,38)(18,30)(20,51,52,50)(25,35,56,55)(27,34)(29,59)(31,45,58,44)(33,57)(37,42)(40,62,41,49)(47,60)(53,54)(61,64);(1,3)(2,8)(4,14)(5,15)(6,17)(7,18)(9,29)(10,30)(11,31)(12,32)(13,23)(16,24)(19,49)(20,40)(21,42)(22,43)(25,44)(26,58)(27,47)(28,48)(33,60)(34,61)(35,46)(36,62)(37,54)(38,55)(39,51)(41,52)(45,56)(50,63)(53,64)(57,59);(1,4,7,22)(2,9,12,34)(3,14,18,43)(5,20,24,52)(6,27,28,21)(8,29,32,61)(10,33,23,54)(11,36,26,19)(13,37,30,60)(15,40,16,41)(17,47,48,42)(25,57,56,53)(31,62,58,49)(35,51,55,50)(38,63,46,39)(44,59,45,64);(3,15)(5,24)(8,30)(10,23)(13,32)(14,40)(16,18)(17,44)(20,52)(25,56)(29,60)(31,46)(33,54)(35,55)(37,61)(38,58)(39,62)(41,43)(42,64)(45,48)(47,59)(49,63)(50,51)(53,57)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=de,e^b=ei,f^c=fi,g^c=ghi,e^d=ei,g^f=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72312431, #72312493, #72312496, … |
| maximal normal subgroup of | #72312431, #72312493, #72312496, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404531);