Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437464 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2403 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5947 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)(5,10)(6,23)(7,12)(8,16)(11,26)(14,41)(15,40)(17,37)(18,31)(19,32)(20,51)(21,38)(22,36)(24,29)(25,35)(27,52)(28,47)(30,44)(33,50)(34,45)(39,43)(42,61)(46,55)(48,57)(49,59)(53,58)(54,63)(56,64)(60,62);(1,3)(2,8)(4,14)(5,15)(6,24)(7,17)(9,30)(10,34)(11,37)(12,29)(13,23)(16,26)(18,42)(19,43)(20,52)(21,47)(22,46)(25,45)(27,32)(28,36)(31,54)(33,39)(35,55)(38,40)(41,51)(44,50)(48,62)(49,64)(53,63)(56,57)(58,60)(59,61);(1,4)(2,9)(3,14)(5,18)(6,20)(7,19)(8,27)(10,31)(11,33)(12,32)(13,39)(15,42)(16,44)(17,43)(21,49)(22,48)(23,51)(24,52)(25,53)(26,50)(28,54)(29,30)(34,56)(35,58)(36,57)(37,41)(38,59)(40,60)(45,63)(46,62)(47,64)(55,61);(1,5,7,22)(2,10,12,36)(3,15,17,46)(4,18,19,48)(6,25,26,21)(8,28,29,34)(9,31,32,57)(11,38,23,35)(13,40,37,55)(14,42,43,62)(16,47,24,45)(20,53,50,49)(27,54,30,56)(33,59,51,58)(39,60,41,61)(44,64,52,63);(4,19)(9,32)(14,43)(18,48)(20,50)(27,30)(31,57)(33,51)(39,41)(42,62)(44,52)(49,53)(54,56)(58,59)(60,61)(63,64)Polycyclic presentation:
2^9|d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,e^c=ei,f^d=fi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72303070, #72303071, #72303085, … |
| maximal normal subgroup of | #72303070, #72303071, #72303085, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344661);