Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500262 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24058 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #61663 | 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 | |
829 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)(3,14)(4,19)(5,10)(6,27)(7,13)(8,16)(9,21)(11,25)(12,29)(15,43)(17,48)(18,38)(20,51)(22,34)(23,50)(24,32)(26,39)(28,42)(30,57)(31,58)(33,55)(35,44)(36,54)(37,59)(40,47)(41,45)(46,60)(49,53)(52,61)(56,62)(63,64);(1,3,7,18)(2,8,13,32)(4,20,21,52)(5,24,25,16)(6,28,29,17)(9,33,19,49)(10,14,11,38)(12,40,27,31)(15,44,45,23)(22,55,54,53)(26,47,59,58)(30,50,60,35)(34,51,36,61)(37,48,39,42)(41,62,43,63)(46,56,57,64);(1,4,5,22,7,21,25,54)(2,9,11,34,13,19,10,36)(3,15,16,46,18,45,24,57)(6,23,26,56,29,44,59,64)(8,30,14,41,32,60,38,43)(12,35,39,62,27,50,37,63)(17,20,47,53,28,52,58,55)(31,33,42,61,40,49,48,51);(2,10)(4,21)(5,25)(8,14)(9,34)(11,13)(12,37)(15,45)(16,24)(19,36)(20,52)(23,44)(26,59)(27,39)(30,43)(31,42)(32,38)(33,51)(35,62)(40,48)(41,60)(47,58)(49,61)(50,63)Polycyclic presentation:
2^9|b^2=i,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,e^b=ei,f^b=fh,g^b=gi,d^c=dh,f^c=fi,h^c=hi,e^d=ei,g^f=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407460);