Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622904 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C25 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
3630 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)(3,14)(4,20)(5,11)(6,26)(7,30)(8,13)(9,18)(10,23)(12,29)(15,44)(16,43)(17,40)(19,51)(21,37)(22,56)(24,38)(25,41)(27,34)(28,54)(31,39)(32,59)(33,49)(35,52)(36,55)(42,48)(45,62)(46,63)(47,53)(50,61)(57,58)(60,64);(1,3)(2,9)(4,15)(5,16)(6,27)(7,19)(8,17)(10,32)(11,33)(12,40)(13,34)(14,26)(18,29)(20,53)(21,45)(22,50)(23,48)(24,46)(25,52)(28,47)(30,35)(31,49)(36,62)(37,60)(38,61)(39,51)(41,43)(42,58)(44,56)(54,63)(55,59)(57,64);(1,4,8,24)(2,10,13,37)(3,15,17,46)(5,23,7,21)(6,28,29,22)(9,32,34,60)(11,38,30,20)(12,36,26,58)(14,42,40,62)(16,48,19,45)(18,50,27,47)(25,55,31,57)(33,61,35,53)(39,56,41,54)(43,63,51,44)(49,64,52,59);(1,5,8,7)(2,11,13,30)(3,16,17,19)(4,21,24,23)(6,25,29,31)(9,33,34,35)(10,20,37,38)(12,39,26,41)(14,43,40,51)(15,45,46,48)(18,49,27,52)(22,55,28,57)(32,53,60,61)(36,54,58,56)(42,44,62,63)(47,64,50,59);(3,17)(9,34)(14,40)(15,46)(16,19)(18,27)(32,60)(33,35)(42,62)(43,51)(44,63)(45,48)(47,50)(49,52)(53,61)(59,64)Polycyclic presentation:
2^9|c^2=i,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,e^b=ei,f^b=fi,g^b=ghi,d^c=di,f^c=fh,g^c=gi,g^f=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72267974, #72267975, #72267976, … |
| maximal normal subgroup of | #72267974, #72267975, #72267976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530102);