Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500748 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2668 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2651 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,13)(4,9)(5,21)(6,25)(7,12)(8,16)(10,23)(11,28)(14,42)(15,46)(17,38)(18,49)(19,34)(20,54)(22,36)(24,39)(26,31)(27,37)(29,55)(30,47)(32,45)(33,52)(35,53)(40,44)(41,48)(43,57)(50,59)(51,61)(56,60)(58,63)(62,64);(1,3)(2,8)(4,14)(5,22)(6,26)(7,17)(9,32)(10,36)(11,38)(12,31)(13,25)(15,23)(16,28)(18,50)(19,44)(20,55)(21,30)(24,48)(27,47)(29,34)(33,59)(35,40)(37,41)(39,46)(42,54)(43,52)(45,53)(49,58)(51,64)(56,63)(57,61)(60,62);(1,4)(2,9)(3,14)(5,18)(6,20)(7,19)(8,29)(10,33)(11,35)(12,34)(13,40)(15,43)(16,45)(17,44)(21,49)(22,50)(23,52)(24,51)(25,54)(26,55)(27,56)(28,53)(30,57)(31,32)(36,62)(37,60)(38,42)(39,61)(41,59)(46,58)(47,63)(48,64);(1,5,7,24)(2,10,12,37)(3,15,17,47)(4,18,19,51)(6,27,28,23)(8,30,31,46)(9,33,34,60)(11,39,25,21)(13,41,38,36)(14,43,44,63)(16,48,26,22)(20,56,53,52)(29,57,32,58)(35,61,54,49)(40,59,42,62)(45,64,55,50);(4,19)(9,34)(14,44)(18,51)(20,53)(29,32)(33,60)(35,54)(40,42)(43,63)(45,55)(49,61)(50,64)(52,56)(57,58)(59,62)Polycyclic presentation:
2^9|d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfg,f^a=fi,h^a=hi,c^b=ch,d^b=df,f^b=fi,g^b=gi,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72306892, #72307027, #72307052, … |
| maximal normal subgroup of | #72306892, #72307027, #72307052, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407945);