Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C22⋊C4 |
| commutator | C2×C4 | C2×C32 |
| Frattini, cosocle | C22×C32 | C22 |
| socle | C22 | C22⋊C32 |
| prime radical | C2×#41 | C16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C22×C32 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 136 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 136 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,20,14,21,36,55,15,22,37,56,40,59,83,104,16,23,38,57,41,60,84,105,43,62,86,107,88,109,124,132,17,24,39,58,42,61,85,106,44,63,87,108,89,110,125,133,12,19,32,51,33,52,77,99,34,53,78,100,79,101,120,130)(11,25,27,64,28,65,70,111,29,66,71,112,72,113,117,134,26,49,67,54,68,95,114,102,69,96,115,103,116,128,135,131,30,50,35,80,73,97,81,121,74,98,82,122,118,129,123,136,31,18,45,46,75,47,90,92,76,48,91,93,119,94,126,127);(1,3)(2,5)(4,8)(6,7)(9,11)(10,18)(12,31)(13,35)(14,28)(15,29)(16,26)(17,30)(19,50)(20,54)(21,47)(22,48)(23,25)(24,49)(27,39)(32,67)(33,75)(34,76)(36,81)(37,82)(38,45)(40,72)(41,68)(42,73)(43,69)(44,74)(46,58)(51,64)(52,97)(53,98)(55,102)(56,103)(57,80)(59,94)(60,65)(61,95)(62,66)(63,96)(70,85)(71,87)(77,114)(78,115)(79,119)(83,123)(84,90)(86,91)(88,116)(89,118)(92,106)(93,108)(99,111)(100,112)(101,129)(104,131)(105,121)(107,122)(109,113)(110,128)(117,125)(120,135)(124,126)(127,133)(130,134)(132,136)Polycyclic presentation:
2^9|a^2=d,c^2=i,d^2=f,f^2=g,g^2=h,h^2=i,b^a=bc,c^a=ce,c^b=ci,d^b=dei152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1707);