Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C32 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C32 | C22⋊C4 |
| commutator | C22 | C4×C32 |
| Frattini, cosocle | C22×C32 | C22 |
| socle | C2 | #7465 |
| prime radical | C23 | C2×C32 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,2,4,10,5,12,21,37,6,13,22,38,25,43,59,81,7,14,23,39,26,44,60,82,28,45,62,83,65,87,101,116,8,11,24,36,27,40,61,78,29,41,63,79,66,84,102,113,30,42,64,80,67,85,103,114,68,86,104,115,105,117,124,127)(3,15,17,46,18,47,53,88,19,48,54,89,56,91,97,118,20,49,55,90,57,92,98,119,58,93,99,120,100,121,123,128,16,9,31,32,50,33,69,72,51,34,70,73,94,75,106,109,52,35,71,74,95,76,107,110,96,77,108,111,122,112,125,126);(1,3)(2,9,11,15)(4,17)(5,18)(6,19)(7,20)(8,16)(10,32,36,46)(12,33,40,47)(13,34,41,48)(14,35,42,49)(21,53)(22,54)(23,55)(24,31)(25,56)(26,57)(27,50)(28,58)(29,51)(30,52)(37,72,78,88)(38,73,79,89)(39,74,80,90)(43,75,84,91)(44,76,85,92)(45,77,86,93)(59,97)(60,98)(61,69)(62,99)(63,70)(64,71)(65,100)(66,94)(67,95)(68,96)(81,109,113,118)(82,110,114,119)(83,111,115,120)(87,112,117,121)(101,123)(102,106)(103,107)(104,108)(105,122)(116,126,127,128)(124,125)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=f,f^2=g,g^2=h,h^2=i,b^a=bc,e^a=ei,c^b=ci176 conjugacy classes — view the full character table.
| maximal subgroup of | #72321570, #72321597, #72321599, … |
| maximal normal subgroup of | #72321570, #72321597, #72321599, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1955);