Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623604 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(1,2)(3,4)(5,6)(7,18)(8,14)(9,25)(10,29)(11,34)(12,16)(13,21)(15,23)(17,32)(19,49)(20,53)(22,51)(24,45)(26,61)(27,64)(28,41)(30,37)(31,47)(33,48)(35,46)(36,43)(38,63)(39,58)(40,62)(42,56)(44,54)(50,55)(52,60)(57,66)(59,67)(65,68);(5,7,12,22)(6,13,16,39)(8,19,11,35)(9,26,28,63)(10,30,33,62)(14,37,34,40)(15,42,43,53)(17,46,45,49)(18,48,51,29)(20,54,56,27)(21,24,58,32)(23,59,36,57)(25,60,41,50)(31,52,65,55)(38,64,61,44)(47,67,68,66);(2,4)(5,8,12,11)(6,14,16,34)(7,19,22,35)(9,23,28,36)(10,24,33,32)(13,37,39,40)(15,41,43,25)(17,29,45,48)(18,49,51,46)(20,52,56,55)(21,30,58,62)(26,59,63,57)(27,31,54,65)(38,66,61,67)(42,50,53,60)(44,68,64,47);(5,9,12,28)(6,15,16,43)(7,20,22,56)(8,23,11,36)(10,31,33,65)(13,38,39,61)(14,41,34,25)(17,47,45,68)(18,50,51,60)(19,52,35,55)(21,57,58,59)(24,54,32,27)(26,62,63,30)(29,44,48,64)(37,66,40,67)(42,49,53,46);(6,16)(13,39)(14,34)(15,43)(17,45)(18,51)(25,41)(29,48)(37,40)(38,61)(42,53)(44,64)(46,49)(47,68)(50,60)(66,67)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,f^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fh,h^a=hi,d^b=df,f^b=fh,h^b=hi,f^d=fh,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530805);