Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396107 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23402 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2625 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,15)(8,19)(9,22)(10,14)(11,17)(12,21)(13,24)(16,30)(18,28)(20,27)(23,25)(26,35)(29,32)(31,33)(34,36);(5,7,10,18)(6,11,14,25)(8,16,21,32)(9,23,24,17)(12,26,19,33)(13,28,22,15)(20,35,36,31)(27,30,34,29);(2,4)(9,24)(11,25)(13,22)(17,23)(20,36)(26,33)(27,34)(31,35);(5,8)(6,12,14,19)(7,16)(9,20)(10,21)(11,26,25,33)(13,27,22,34)(15,29,28,30)(17,31)(18,32)(23,35)(24,36)Polycyclic presentation:
2^9|b^2=i,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72261630, #72306319, #72306507, … |
| maximal normal subgroup of | #72261630, #72306319, #72306507, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311182);