Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#120 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(5,6)(7,12)(8,19)(9,14)(10,22)(11,15)(13,20)(16,32)(17,26)(18,27)(21,30)(23,28)(24,31)(25,35)(29,34)(33,36);(1,2)(3,4)(5,7)(6,12)(8,16)(9,18)(10,23)(11,17)(13,29)(14,27)(15,26)(19,32)(20,34)(21,33)(22,28)(24,35)(25,31)(30,36);(1,3,4,2)(5,8)(6,13)(7,16)(9,20)(10,24)(11,21)(12,25)(14,30)(15,31)(17,33)(18,34)(19,22)(23,35)(26,29)(27,32)(28,36);(7,17)(12,26)(16,33)(18,23)(25,29)(27,28)(32,36)(34,35);(5,9,11,10)(6,14,15,22)(7,18,17,23)(8,20,21,24)(12,27,26,28)(13,30,31,19)(16,34,33,35)(25,32,29,36)Polycyclic presentation:
2^9|c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,h^a=hi,c^b=ch,d^b=di,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72312497, #72312550, #72312566, … |
| maximal normal subgroup of | #72312497, #72312550, #72312566, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339077);