Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623626 |
| 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 | |
1940 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)(12,22)(16,28)(17,29)(19,44)(20,43)(21,33)(23,51)(24,52)(25,36)(26,37)(30,40)(31,42)(32,47)(34,49)(35,50)(38,53)(39,54)(41,57)(45,64)(46,58)(48,67)(55,61)(56,63)(59,65)(60,66)(62,68);(5,7)(6,12)(8,19)(9,20)(10,31)(11,21)(13,32)(14,34)(15,35)(16,42)(17,43)(18,27)(22,28)(23,29)(24,45)(25,56)(26,46)(30,49)(33,40)(36,59)(37,60)(38,63)(39,64)(41,51)(44,52)(47,53)(48,54)(50,57)(55,65)(58,61)(62,67)(66,68);(2,4)(5,8,11,26)(6,13,15,37)(7,19,21,46)(9,24,30,55)(10,25,29,54)(12,32,35,60)(14,36,17,39)(16,38,41,62)(18,44,33,58)(20,45,49,65)(22,47,50,66)(23,48,31,56)(27,52,40,61)(28,53,57,68)(34,59,43,64)(42,63,51,67);(6,14)(7,20)(9,28)(10,29)(12,33)(13,36)(15,17)(18,35)(19,45)(21,49)(22,23)(24,53)(25,54)(27,40)(30,57)(31,50)(32,58)(34,42)(37,39)(43,51)(44,60)(46,65)(47,48)(52,61)(55,68)(56,66)(59,63)(64,67);(6,15)(7,21)(13,37)(14,17)(16,41)(19,46)(20,49)(22,50)(23,31)(27,40)(36,39)(38,62)(45,65)(47,66)(48,56)(52,61)Polycyclic presentation:
2^9|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,e^b=ei,f^b=fh,h^b=hi,f^d=fh,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72343374, #72343827, #72343839 |
| maximal normal subgroup of | #72343374, #72343827, #72343839 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530824);