Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #214656 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3486 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3770 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)(5,6,9,14,10,13,24,28)(7,15,18,29,17,11,22,26)(8,19,20,30,21,12,23,27)(16,25,33,35,32,31,34,36);(2,4)(5,7)(6,11)(8,16)(9,22)(10,17)(12,25)(13,15)(14,29)(18,24)(19,31)(20,34)(21,32)(23,33)(26,28)(27,36)(30,35);(11,15)(25,31)(26,29)(35,36);(2,4)(5,8,10,21)(6,12,13,19)(7,16,17,32)(9,23,24,20)(11,25,15,31)(14,30,28,27)(18,34,22,33)(26,36,29,35)Polycyclic presentation:
2^9|a^2=h,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^b=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72303273, #72303351, #72318058, … |
| maximal normal subgroup of | #72303273, #72303351, #72318058, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266790);