Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153082 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2218 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13515 |
| prime radical | C23 | C2×Q32 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2,5,11,27,44,73,65,7,14,29,47,33,48,81,102,8,12,30,45,79,101,123,118,32,49,80,103,6,13,28,46)(3,9,16,36,57,91,108,106,19,38,60,93,61,97,113,128,17,39,58,94,112,126,122,119,62,96,114,127,18,37,59,92)(4,20,23,50,31,83,78,42,25,67,74,99,72,100,76,40,22,66,71,105,85,104,82,51,34,86,77,41,24,43,26,10)(15,52,53,95,63,116,111,89,55,107,109,125,70,121,75,87,21,68,69,120,84,124,115,98,64,117,110,88,54,90,56,35);(1,3)(2,9)(4,21,22,15)(5,16)(6,18)(7,19)(8,17)(10,35,40,87)(11,36)(12,39)(13,37)(14,38)(20,52,66,68)(23,69,71,53)(24,70,72,54)(25,64,34,55)(26,75,76,56)(27,57)(28,59)(29,60)(30,58)(31,84,85,63)(32,62)(33,61)(41,88,99,125)(42,89,51,98)(43,90,100,121)(44,91)(45,94)(46,92)(47,93)(48,97)(49,96)(50,95,105,120)(65,106)(67,107,86,117)(73,108)(74,110,77,109)(78,115,82,111)(79,112)(80,114)(81,113)(83,116,104,124)(101,126)(102,128)(103,127)(118,119)(122,123);(1,4,7,34,8,22,32,25)(2,10,14,51,12,40,49,42)(3,15,19,64,17,21,62,55)(5,26,29,82,30,76,80,78)(6,31,27,24,33,85,79,72)(9,35,38,98,39,87,96,89)(11,43,47,104,45,100,103,83)(13,50,44,41,48,105,101,99)(16,56,60,115,58,75,114,111)(18,63,57,54,61,84,112,70)(20,65,86,102,66,118,67,46)(23,73,77,81,71,123,74,28)(36,90,93,124,94,121,127,116)(37,95,91,88,97,120,126,125)(52,106,117,128,68,119,107,92)(53,108,110,113,69,122,109,59)Polycyclic presentation:
2^9|a^2=e,b^2=d,c^2=dh,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=eg,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72320516, #72320849, #72320880 |
| maximal normal subgroup of | #72320516, #72320849, #72320880 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60590);