Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7548 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1424 |
| commutator | C32 | C22×C4 |
| Frattini, cosocle | C2×C32 | C23 |
| socle | C2 | C2×#1424 |
| prime radical | C23 | SD64 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2)(3,16,20,55)(4,10)(5,27)(6,32)(7,35)(8,15)(9,19,39,69)(11,29)(12,33)(13,25)(14,26)(17,43)(18,53,65,58)(21,74,72,110)(22,76,73,108)(23,77)(24,81)(28,51)(30,90)(31,87)(34,46)(36,45)(37,80)(38,64,42,103)(40,100,98,125)(41,102,99,127)(44,82)(47,62)(48,63)(49,60)(50,78)(52,84)(54,109,91,70)(56,68,112,75)(57,114,113,67)(59,106,115,111)(61,119)(66,123,122,126)(71,97,116,93)(79,86)(83,92)(85,89)(88,104)(94,124,107,121)(95,120)(96,117,101,128)(105,118);(1,3,17,59,6,21,61,116,25,67,88,125,82,124,31,69,4,18,60,109,24,66,118,127,36,73,120,128,28,68,80,103,8,20,63,115,34,72,30,71,84,114,89,100,78,121,5,19,26,65,29,70,83,122,79,102,7,22,62,117,33,75,23,64)(2,9,37,94,12,40,95,113,35,93,105,110,92,106,49,55,14,42,87,112,50,101,104,108,52,99,119,123,46,54,43,53,15,39,77,107,51,98,47,57,45,97,86,74,81,111,11,16,10,38,27,56,44,96,85,76,13,41,90,126,32,91,48,58);(1,4,8,26)(2,10,15,14)(3,18,20,65)(5,23,31,80)(6,24,34,83)(7,25,36,84)(9,38,39,42)(11,43,49,48)(12,44,51,50)(13,45,52,35)(16,53,55,58)(17,60,63,29)(19,64,69,103)(21,66,72,122)(22,67,73,114)(27,77,87,37)(28,78,33,82)(30,79,61,118)(32,81,46,92)(40,96,98,101)(41,97,99,93)(47,104,95,85)(54,106,91,111)(56,107,112,94)(57,108,113,76)(59,109,115,70)(62,88,120,89)(68,121,75,124)(71,102,116,127)(74,123,110,126)(86,119,105,90)(100,117,125,128)Polycyclic presentation:
2^9|a^2=e,b^2=dfh,c^2=i,d^2=fgi,f^2=gh,g^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fg,g^a=gh,h^a=hi,e^b=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72321113, #72321194, #72321207, … |
| maximal normal subgroup of | #72321113, #72321194, #72321207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60774);