Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #13742 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D32 |
| commutator | C32 | C22×C4 |
| Frattini, cosocle | C2×C32 | C23 |
| prime radical | C23 | D32 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 132 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 132 points:
(1,2,3,4)(5,6)(7,20)(8,15)(9,30)(10,37)(11,42)(12,45)(13,19)(14,22)(16,33)(17,32)(18,39)(21,61)(23,71)(24,75)(25,77)(26,63)(27,78)(28,83)(29,59)(31,56)(34,95)(35,98)(36,60)(38,53)(40,103)(41,100)(43,62)(44,104)(46,64)(47,89)(48,88)(49,70)(50,69)(51,81)(52,80)(54,115)(55,67)(57,72)(58,68)(65,116)(66,123)(73,129)(74,128)(76,130)(79,112)(82,121)(84,111)(85,91)(86,93)(87,118)(90,119)(92,99)(94,105)(96,109)(97,122)(101,113)(102,114)(106,132)(107,131)(108,124)(110,127)(117,126)(120,125);(5,7,13,26)(6,14,19,49)(8,25,12,21)(9,31,36,90)(10,38,41,102)(11,43,44,66)(15,50,45,46)(16,53,55,114)(17,56,57,119)(18,58,59,110)(20,60,63,30)(22,67,70,33)(23,72,74,32)(24,29,76,39)(27,79,35,87)(28,84,40,101)(34,96,97,132)(37,47,100,107)(42,48,104,108)(51,111,105,113)(52,112,99,118)(54,116,117,129)(61,78,77,98)(62,122,123,95)(64,81,69,94)(65,80,73,92)(68,126,127,115)(71,120,128,85)(75,121,130,86)(82,131,93,89)(83,109,103,106)(88,125,124,91);(5,8,13,12)(6,15,19,45)(7,21,26,25)(9,27,36,35)(10,28,41,40)(11,29,44,39)(14,46,49,50)(16,51,55,105)(17,52,57,99)(18,42,59,104)(20,61,63,77)(22,64,70,69)(23,65,74,73)(24,66,76,43)(30,78,60,98)(31,87,90,79)(32,80,72,92)(33,81,67,94)(34,82,97,93)(37,83,100,103)(38,101,102,84)(47,106,107,109)(48,58,108,110)(53,113,114,111)(54,85,117,120)(56,118,119,112)(62,75,123,130)(68,124,127,88)(71,116,128,129)(86,95,121,122)(89,132,131,96)(91,126,125,115)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=i,d^2=fg,f^2=gh,g^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fg,g^a=gh,h^a=hi,c^b=ci,d^b=df,f^b=fg,g^b=gh,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60754);