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 | |
1948 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)(9,27)(10,14)(11,15)(12,21)(16,28)(17,29)(19,44)(20,43)(22,52)(23,33)(24,53)(25,36)(26,37)(30,40)(31,42)(32,46)(34,49)(35,50)(38,54)(39,55)(41,51)(45,64)(47,68)(48,58)(56,61)(57,63)(59,65)(60,66)(62,67);(5,7,11,23)(6,12,15,34)(8,19,26,48)(9,20,30,50)(10,31,29,22)(13,32,37,59)(14,35,17,43)(16,42,41,52)(18,40,33,27)(21,51,49,28)(24,45,56,66)(25,57,55,47)(36,60,39,64)(38,63,62,68)(44,61,58,53)(46,67,65,54);(2,4)(5,8,11,26)(6,13,15,37)(7,19,23,48)(9,24,30,56)(10,25,29,55)(12,32,34,59)(14,36,17,39)(16,38,41,62)(18,44,33,58)(20,45,50,66)(21,46,49,65)(22,47,31,57)(27,53,40,61)(28,54,51,67)(35,60,43,64)(42,63,52,68);(6,14)(7,20)(9,28)(10,29)(12,33)(13,36)(15,17)(18,34)(19,45)(21,22)(23,50)(24,54)(25,55)(27,40)(30,51)(31,49)(32,58)(35,42)(37,39)(43,52)(44,59)(46,47)(48,66)(53,61)(56,67)(57,65)(60,63)(64,68);(6,15)(12,34)(13,37)(14,17)(16,41)(18,33)(27,40)(32,59)(35,43)(36,39)(38,62)(42,52)(44,58)(53,61)(60,64)(63,68)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,d^b=df,f^b=fh,h^b=hi,f^d=fh,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72343374, #72343825, #72343835 |
| maximal normal subgroup of | #72343374, #72343825, #72343835 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530803);