Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499825 |
| 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 |
| prime radical | #60401 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1853 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 1853 conjugacy classes of subgroups — too many to draw. Only its 148 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(6,12)(7,29)(8,14)(9,18)(11,25)(16,48)(17,32)(19,46)(20,55)(21,38)(22,37)(23,44)(24,39)(26,59)(27,42)(28,41)(30,47)(31,53)(33,61)(34,54)(35,56)(36,57)(40,58)(43,52)(45,50)(49,63)(51,62)(60,64);(1,3,8,19)(2,9,14,34)(4,20,24,51)(5,17,7,30)(6,18,27,54)(10,35,39,64)(11,32,41,47)(12,15,42,46)(13,43,29,33)(16,21,50,23)(22,56,26,60)(25,52,28,61)(31,44,63,38)(36,45,40,48)(37,55,59,62)(49,57,53,58);(1,4,8,24)(2,10,14,39)(3,16,19,50)(5,23,7,21)(6,26,27,22)(9,31,34,63)(11,40,41,36)(12,37,42,59)(13,44,29,38)(15,45,46,48)(17,51,30,20)(18,53,54,49)(25,57,28,58)(32,55,47,62)(33,64,43,35)(52,56,61,60);(1,5,8,7)(2,11,14,41)(3,17,19,30)(4,21,24,23)(6,25,27,28)(9,32,34,47)(10,36,39,40)(12,13,42,29)(15,43,46,33)(16,20,50,51)(18,52,54,61)(22,57,26,58)(31,62,63,55)(35,45,64,48)(37,38,59,44)(49,56,53,60)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,h^a=hi,c^b=cg,f^b=fh,g^b=gi,d^c=di,e^c=eh,g^c=gi,f^d=fhi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72281882, #72282045, #72282134, … |
| maximal normal subgroup of | #72281882, #72282045, #72282134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407026);