Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623665 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×D4×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#443 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,9,15)(3,16,19,50)(4,22,11,36)(5,25,28,24)(6,29,34,45)(7,14,8,13)(10,37,39,20)(12,32,46,33)(17,51,41,18)(21,52,35,43)(23,58,42,27)(26,59,61,38)(30,47,62,40)(31,44,63,60)(48,54,55,49)(53,57,64,56);(1,3)(2,10)(4,17)(5,26)(6,30)(7,21)(8,35)(9,19)(11,41)(12,43)(13,40)(14,47)(15,39)(16,29)(18,27)(20,33)(22,57)(23,59)(24,49)(25,54)(28,61)(31,48)(32,37)(34,62)(36,56)(38,42)(44,64)(45,50)(46,52)(51,58)(53,60)(55,63);(1,4,9,11)(2,5,15,28)(3,17,19,41)(6,23,34,42)(7,25,8,24)(10,26,39,61)(12,27,46,58)(13,36,14,22)(16,48,50,55)(18,52,51,43)(20,53,37,64)(21,54,35,49)(29,31,45,63)(30,59,62,38)(32,44,33,60)(40,56,47,57);(1,5)(2,11)(3,18)(4,15)(6,31)(7,22)(8,36)(9,28)(10,38)(12,44)(13,25)(14,24)(16,49)(17,52)(19,51)(20,56)(21,48)(23,45)(26,30)(27,33)(29,42)(32,58)(34,63)(35,55)(37,57)(39,59)(40,64)(41,43)(46,60)(47,53)(50,54)(61,62);(3,19)(10,39)(16,50)(17,41)(18,51)(20,37)(21,35)(26,61)(30,62)(38,59)(40,47)(43,52)(48,55)(49,54)(53,64)(56,57)Polycyclic presentation:
2^9|a^2=i,c^2=i,f^2=hi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fh,g^a=gi,h^a=hi,d^b=df,e^b=ei,f^b=fh,h^b=hi,g^c=gi,f^d=fh,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530863);