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) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C32 | C2×D4 |
| commutator | C4 | C22×C32 |
| Frattini, cosocle | C2×C32 | C23 |
| socle | C2 | D4×C32 |
| prime radical | #135 | C16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C2×C64 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2,6,14,7,15,36,57,8,16,37,58,40,60,93,110,9,13,38,54,41,55,94,108,43,56,73,98,96,109,117,125,10,17,39,59,42,61,95,111,44,62,63,105,97,112,113,128,5,31,33,86,34,87,91,123,35,77,46,100,92,119,101,126)(3,18,20,64,21,65,68,114,22,25,51,78,70,79,107,120,19,11,45,47,66,48,99,102,67,4,24,26,75,27,76,80,23,28,69,81,71,83,115,121,72,29,74,82,116,84,118,122,32,49,88,103,89,104,124,127,90,30,12,50,52,85,53,106);(1,3)(2,11)(4,16)(5,32)(6,20)(7,21)(8,22)(9,19)(10,23)(12,46)(13,28)(14,47)(15,48)(17,49)(18,31)(24,73)(25,77)(26,58)(27,60)(29,56)(30,62)(33,88)(34,89)(35,90)(36,68)(37,51)(38,45)(39,69)(40,70)(41,66)(42,71)(43,67)(44,72)(50,105)(52,92)(53,101)(54,81)(55,83)(57,102)(59,103)(61,104)(63,74)(64,86)(65,87)(75,96)(76,117)(78,100)(79,119)(80,110)(82,98)(84,109)(85,112)(91,124)(93,107)(94,99)(95,115)(97,116)(106,128)(108,121)(111,127)(113,118)(114,123)(120,126)(122,125);(1,4,6,26,7,27,36,80,8,28,37,81,40,83,93,121,9,29,38,82,41,84,94,122,43,49,73,103,96,104,117,127,10,30,39,50,42,85,95,106,44,18,63,64,97,65,113,114,5,25,33,78,34,79,91,120,35,11,46,47,92,48,101,102)(2,12,14,52,15,53,57,3,16,20,58,21,60,68,110,22,13,51,54,70,55,107,108,19,56,45,98,66,109,99,125,67,17,24,59,75,61,76,111,23,62,69,105,71,112,115,128,72,31,74,86,116,87,118,123,32,77,88,100,89,119,124,126,90)Polycyclic presentation:
2^9|a^2=e,c^2=e,d^2=i,e^2=f,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ci,d^a=di,d^b=di176 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, 60783);