Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622907 |
| 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 ◁ C2×Q8 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
3631 conjugacy classes of subgroups; only the 568 normal subgroups are shown.
This group has 3631 conjugacy classes of subgroups — too many to draw. Only its 568 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,15)(4,21)(5,12)(6,13)(7,27)(8,30)(9,14)(10,18)(11,22)(16,44)(17,43)(19,51)(20,34)(23,54)(24,37)(25,40)(26,41)(28,36)(29,55)(31,39)(32,59)(33,50)(35,49)(38,56)(42,45)(46,60)(47,53)(48,63)(52,61)(57,58)(62,64);(1,3)(2,10)(4,16)(5,17)(6,18)(7,28)(8,19)(9,20)(11,32)(12,33)(13,34)(14,36)(15,27)(21,53)(22,45)(23,46)(24,48)(25,49)(26,50)(29,47)(30,35)(31,52)(37,60)(38,61)(39,62)(40,43)(41,51)(42,58)(44,54)(55,63)(56,59)(57,64);(1,4)(2,11)(3,16)(5,22)(6,23)(7,29)(8,31)(9,24)(10,32)(12,37)(13,38)(14,39)(15,42)(17,45)(18,46)(19,52)(20,48)(21,30)(25,56)(26,57)(27,58)(28,47)(33,60)(34,61)(35,53)(36,62)(40,55)(41,54)(43,63)(44,51)(49,59)(50,64);(1,5,9,8)(2,12,14,30)(3,17,20,19)(4,22,24,31)(6,26,7,25)(10,33,36,35)(11,37,39,21)(13,41,27,40)(15,43,34,51)(16,45,48,52)(18,50,28,49)(23,57,29,56)(32,60,62,53)(38,54,58,55)(42,63,61,44)(46,64,47,59);(1,6,9,7)(2,13,14,27)(3,18,20,28)(4,23,24,29)(5,25,8,26)(10,34,36,15)(11,38,39,58)(12,40,30,41)(16,46,48,47)(17,49,19,50)(21,54,37,55)(22,56,31,57)(32,61,62,42)(33,43,35,51)(44,60,63,53)(45,59,52,64)Polycyclic presentation:
2^9|d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,f^b=fi,g^b=ghi,f^c=fh,g^c=gi,e^d=ei,g^f=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72209895, #72209896, #72304193 |
| maximal normal subgroup of | #72209895, #72209896, #72304193 |
| maximal quotient of | #72227698, #72227830, #72228008, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530105);