Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623150 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#3109 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | #33530 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2873 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,13,15,42)(4,18,19,51)(5,22,23,56)(6,28,26,11)(8,30,32,16)(9,34,35,21)(10,37,38,27)(14,44,45,64)(17,49,29,39)(20,40,25,36)(24,41,58,33)(31,53,61,59)(43,55,62,47)(46,50,48,63)(52,60,57,54);(5,23)(10,38)(13,42)(16,30)(20,25)(22,56)(24,58)(27,37)(36,40)(39,49)(43,62)(44,64)(46,48)(52,57)(53,59)(54,60);(1,3)(2,8)(4,14)(5,24)(6,29)(7,15)(9,31)(10,39)(11,41)(12,32)(13,22)(16,27)(17,26)(18,50)(19,45)(20,53)(21,55)(23,58)(25,59)(28,33)(30,37)(34,47)(35,61)(36,64)(38,49)(40,44)(42,56)(43,57)(46,54)(48,60)(51,63)(52,62);(1,4,6,21,7,19,26,34)(2,9,11,18,12,35,28,51)(3,14,29,55,15,45,17,47)(5,25,27,60,23,20,37,54)(8,31,41,50,32,61,33,63)(10,40,22,57,38,36,56,52)(13,43,49,64,42,62,39,44)(16,48,58,53,30,46,24,59);(4,19)(9,35)(14,45)(18,51)(20,25)(21,34)(31,61)(36,40)(43,62)(44,64)(46,48)(47,55)(50,63)(52,57)(53,59)(54,60)Polycyclic presentation:
2^9|a^2=i,d^2=fh,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gh,h^a=hi,g^b=gi,f^c=fi,g^c=gh,h^c=hi,e^d=ei,g^d=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530347);