Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C32 | C2×D4 |
| commutator | C4 | C22×C32 |
| Frattini, cosocle | C2×C32 | C23 |
| socle | C2 | C2×C22⋊C32 |
| prime radical | C2×#41 | C16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
180 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2,5,13,6,14,28,50,7,15,29,51,32,53,75,99,8,16,30,52,33,54,76,100,35,55,78,101,80,102,116,125,9,11,31,42,34,43,77,89,36,44,79,90,81,91,117,121,4,12,24,46,25,47,69,94,26,48,70,95,71,96,112,123)(3,17,19,56,20,57,62,103,21,58,63,104,64,105,109,126,18,38,59,49,60,84,106,97,61,85,107,98,108,119,127,124,22,45,27,72,65,92,73,113,66,93,74,114,110,122,115,128,23,10,37,39,67,40,82,86,68,41,83,87,111,88,118,120);(1,3)(2,10)(4,23)(5,27)(6,20)(7,21)(8,18)(9,22)(11,38)(12,45)(13,49)(14,40)(15,41)(16,17)(19,31)(24,59)(25,67)(26,68)(28,73)(29,74)(30,37)(32,64)(33,60)(34,65)(35,61)(36,66)(39,42)(43,84)(44,85)(46,56)(47,92)(48,93)(50,97)(51,98)(52,72)(53,88)(54,57)(55,58)(62,77)(63,79)(69,106)(70,107)(71,111)(75,115)(76,82)(78,83)(80,108)(81,110)(86,89)(87,90)(91,119)(94,103)(95,104)(96,122)(99,124)(100,113)(101,114)(102,105)(109,117)(112,127)(116,118)(120,121)(123,126)(125,128);(2,11)(10,38)(12,16)(13,42)(14,43)(15,44)(17,45)(39,49)(40,84)(41,85)(46,52)(47,54)(48,55)(50,89)(51,90)(53,91)(56,72)(57,92)(58,93)(86,97)(87,98)(88,119)(94,100)(95,101)(96,102)(99,121)(103,113)(104,114)(105,122)(120,124)(123,125)(126,128)Polycyclic presentation:
2^9|a^2=e,d^2=i,e^2=f,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ci,d^b=di,e^b=ei176 conjugacy classes — view the full character table.
| maximal subgroup of | #72321570, #72321604, #72321610, … |
| maximal normal subgroup of | #72321570, #72321604, #72321610, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60628);