Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3192 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)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,24,28,25,27,31,34)(23,26,30,33,29,32,35,36);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18);(2,4)(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,23)(22,26)(24,30)(25,29)(27,32)(28,33)(31,35)(34,36);(6,12)(10,13)(11,20)(18,19)(22,27)(26,32)(28,34)(33,36);(23,29)(26,32)(30,35)(33,36)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=cf,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239870, #72239885, #72239907, … |
| maximal normal subgroup of | #72239870, #72239885, #72239907, … |
| maximal quotient of | #72239872, #72305011, #72314330 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281578);