Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6629045 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2847 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60398 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,13,16,36)(4,9,20,31)(5,10,22,33)(6,23,24,11)(8,25,28,17)(14,34,38,56)(15,35,40,57)(18,42,44,29)(19,45,46,30)(21,47,48,32)(26,49,52,39)(27,50,53,41)(37,58,59,55)(43,54,63,62)(51,60,61,64);(1,3,6,17,7,16,24,25)(2,8,11,13,12,28,23,36)(4,14,19,39,20,38,46,49)(5,15,21,41,22,40,48,50)(9,26,30,34,31,52,45,56)(10,27,32,35,33,53,47,57)(18,37,43,60,44,59,63,64)(29,51,54,55,42,61,62,58);(1,4)(2,9)(3,14)(5,18)(6,19)(7,20)(8,26)(10,29)(11,30)(12,31)(13,34)(15,37)(16,38)(17,39)(21,43)(22,44)(23,45)(24,46)(25,49)(27,51)(28,52)(32,54)(33,42)(35,55)(36,56)(40,59)(41,60)(47,62)(48,63)(50,64)(53,61)(57,58);(1,5)(2,10)(3,15)(4,18)(6,21)(7,22)(8,27)(9,29)(11,32)(12,33)(13,35)(14,37)(16,40)(17,41)(19,43)(20,44)(23,47)(24,48)(25,50)(26,51)(28,53)(30,54)(31,42)(34,55)(36,57)(38,59)(39,60)(45,62)(46,63)(49,64)(52,61)(56,58);(3,16)(8,28)(13,36)(14,38)(15,40)(17,25)(26,52)(27,53)(34,56)(35,57)(37,59)(39,49)(41,50)(51,61)(55,58)(60,64)Polycyclic presentation:
2^9|a^2=i,b^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,e^b=ei,h^c=hi,g^d=gi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72312588, #72312598, #72312651, … |
| maximal normal subgroup of | #72312588, #72312598, #72312651, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536242);