Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | #102 |
| commutator | C8 | C2×C32 |
| Frattini, cosocle | C4×C32 | C22 |
| socle | C2 | D4⋊C32 |
| prime radical | #135 | C16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C2×C32 ◁ C4×C32 ◁ G | |
| LECS | G ▷ C4×C32 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
73 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,2,5,13,7,15,32,55,8,16,33,56,39,60,87,109,9,14,34,54,40,58,88,107,29,59,79,108,81,111,120,128,10,17,35,57,41,61,89,110,27,12,49,51,80,52,102,105,6,36,31,85,38,90,86,123,4,24,26,73,28,74,78,117)(3,18,20,62,22,63,70,112,19,48,65,53,67,101,114,106,21,50,69,103,71,104,116,127,66,44,77,94,115,98,119,124,23,64,30,82,72,113,84,121,68,11,42,43,92,45,93,95,37,46,83,96,91,99,122,125,25,47,75,97,76,100,118,126);(1,3)(2,11)(4,25)(5,30)(6,37)(7,22)(8,19)(9,21)(10,23)(12,50)(13,53)(14,47)(15,45)(16,46)(17,48)(18,59)(20,35)(24,64)(26,77)(27,68)(28,76)(29,66)(31,69)(32,84)(33,42)(34,83)(36,44)(38,91)(39,67)(40,71)(41,72)(43,57)(49,65)(51,96)(52,104)(54,94)(55,106)(56,103)(58,100)(60,99)(61,101)(62,73)(63,111)(70,89)(74,113)(75,79)(78,119)(80,92)(81,115)(82,108)(85,97)(86,116)(87,93)(88,122)(90,98)(95,110)(102,114)(105,125)(107,124)(109,127)(112,117)(118,120)(121,128)(123,126)Polycyclic presentation:
2^9|a^2=d,c^2=e,d^2=f,e^2=i,f^2=g,g^2=h,h^2=i,b^a=bc,c^a=ce,e^a=ei,c^b=cei,d^b=di,e^b=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72230752 |
| maximal normal subgroup of | #72230752 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1802);