Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431739 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#392 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6981 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)(3,15)(4,20)(5,11)(6,13)(7,12)(8,14)(9,18)(10,23)(16,29)(17,41)(19,39)(21,48)(22,35)(24,34)(25,36)(26,38)(27,37)(28,31)(30,46)(32,44)(33,51)(40,43)(42,53)(45,59)(47,54)(49,58)(50,57)(52,55)(56,61)(60,62)(63,64);(1,3)(2,9)(4,16)(5,17)(6,28)(7,18)(8,19)(10,32)(11,30)(12,39)(13,15)(14,31)(20,40)(21,42)(22,47)(23,43)(24,44)(25,45)(26,52)(27,46)(29,35)(33,56)(34,54)(36,55)(37,59)(38,41)(48,60)(49,61)(50,63)(51,62)(53,57)(58,64);(1,4,8,24)(2,10,14,35)(3,16,19,44)(5,21,25,49)(6,23,7,22)(9,29,31,32)(11,33,36,57)(12,20,13,34)(15,40,39,54)(17,42,45,61)(18,47,28,43)(26,51,27,50)(30,53,55,56)(37,48,38,58)(41,60,59,64)(46,63,52,62);(1,5)(2,11)(3,17)(4,21)(6,26)(7,27)(8,25)(9,30)(10,33)(12,37)(13,38)(14,36)(15,41)(16,42)(18,46)(19,45)(20,48)(22,50)(23,51)(24,49)(28,52)(29,53)(31,55)(32,56)(34,58)(35,57)(39,59)(40,60)(43,62)(44,61)(47,63)(54,64);(5,25)(11,36)(17,45)(21,49)(26,27)(30,55)(33,57)(37,38)(41,59)(42,61)(46,52)(48,58)(50,51)(53,56)(60,64)(62,63)Polycyclic presentation:
2^9|c^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,f^b=fi,f^c=fi,g^c=gi,e^d=ei89 conjugacy classes — view the full character table.
| maximal subgroup of | #72307018, #72307021, #72307024, … |
| maximal normal subgroup of | #72307018, #72307021, #72307024, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338936);