Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623613 |
| 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 | |
1930 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)(8,13)(9,28)(10,14)(11,15)(12,21)(16,29)(17,30)(19,46)(20,45)(22,53)(23,34)(24,54)(25,58)(26,38)(27,39)(31,42)(32,44)(33,48)(35,50)(36,51)(37,56)(40,59)(41,60)(43,52)(47,63)(49,66)(55,62)(57,67)(61,64)(65,68);(5,7,11,23)(6,12,15,35)(8,24,27,19)(9,20,31,51)(10,32,30,22)(13,37,39,33)(14,36,17,45)(16,44,43,53)(18,42,34,28)(21,52,50,29)(25,55,61,47)(26,49,60,57)(38,63,41,62)(40,66,65,67)(46,58,54,64)(48,59,56,68);(2,4)(5,8,11,27)(6,13,15,39)(7,19,23,24)(9,25,31,61)(10,26,30,60)(12,33,35,37)(14,38,17,41)(16,40,43,65)(18,46,34,54)(20,47,51,55)(21,48,50,56)(22,49,32,57)(28,58,42,64)(29,59,52,68)(36,62,45,63)(44,67,53,66);(6,14)(7,20)(9,29)(10,30)(12,34)(13,38)(15,17)(18,35)(19,47)(21,22)(23,51)(24,55)(25,59)(26,60)(28,42)(31,52)(32,50)(33,54)(36,44)(37,46)(39,41)(45,53)(48,49)(56,57)(58,64)(61,68)(62,67)(63,66);(6,15)(12,35)(13,39)(14,17)(16,43)(18,34)(28,42)(33,37)(36,45)(38,41)(40,65)(44,53)(46,54)(58,64)(62,63)(66,67)Polycyclic presentation:
2^9|b^2=i,c^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,f^d=fh,h^d=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72343374, #72343827, #72343835 |
| maximal normal subgroup of | #72343374, #72343827, #72343835 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530811);