Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623604 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
2004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,15)(8,20)(9,25)(10,13)(11,18)(12,23)(14,22)(16,33)(17,35)(19,34)(21,31)(24,32)(26,29)(27,30)(28,36);(5,7)(6,11)(8,21)(9,26)(10,19)(12,29)(13,16)(14,31)(15,20)(17,22)(18,23)(24,36)(25,28)(27,33)(30,35)(32,34);(2,4)(5,6)(7,16)(8,12)(9,14)(10,13)(11,19)(15,33)(17,31)(18,34)(20,23)(21,35)(22,25)(24,30)(26,28)(27,32)(29,36);(5,8)(6,12)(7,17)(9,27)(10,24)(11,28)(13,30)(14,32)(15,34)(16,31)(18,33)(19,26)(20,25)(22,23);(6,13)(11,16)(12,30)(14,25)(15,34)(20,32)(28,31)(29,35)Polycyclic presentation:
2^9|f^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fh,h^a=hi,c^b=ci,d^b=df,f^b=fh,h^b=hi,e^c=ei,f^d=fh,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72321856, #72322060, #72322387, … |
| maximal normal subgroup of | #72321856, #72322060, #72322387, … |
| maximal quotient of | #72343836 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530835);