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 ◁ #532 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1920 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 68 points:
(1,2)(3,4)(5,6)(7,17)(8,23)(9,28)(10,34)(11,14)(12,20)(13,26)(15,32)(16,31)(18,48)(19,52)(21,58)(22,50)(24,51)(25,63)(27,46)(29,45)(30,44)(33,47)(35,43)(36,40)(37,62)(38,61)(39,55)(41,53)(42,56)(49,54)(57,66)(59,60)(64,67)(65,68);(5,7,11,22)(6,12,14,18)(8,24,27,62)(9,29,33,61)(10,35,31,21)(13,39,40,52)(15,43,42,58)(16,45,34,38)(17,47,50,28)(19,53,55,25)(20,56,48,32)(23,60,46,49)(26,64,36,57)(30,59,65,54)(37,63,51,41)(44,67,68,66);(2,4)(5,6)(7,18)(8,13)(9,15)(10,16)(11,14)(12,22)(17,48)(19,51)(20,50)(21,45)(23,26)(24,52)(25,41)(27,40)(28,32)(29,58)(30,44)(31,34)(33,42)(35,38)(36,46)(37,55)(39,62)(43,61)(47,56)(49,64)(53,63)(54,67)(57,60)(59,66)(65,68);(5,8,11,27)(6,13,14,40)(7,19,22,55)(9,30,33,65)(10,36,31,26)(12,37,18,51)(15,44,42,68)(16,46,34,23)(17,49,50,60)(20,57,48,64)(21,59,35,54)(24,61,62,29)(25,56,53,32)(28,41,47,63)(38,67,45,66)(39,58,52,43);(6,14)(12,18)(13,40)(15,42)(16,34)(17,50)(23,46)(28,47)(37,51)(38,45)(39,52)(41,63)(43,58)(44,68)(49,60)(66,67)Polycyclic presentation:
2^9|b^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,c^b=ci,d^b=df,f^b=fh,h^b=hi,e^c=ei,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, 7530837);