Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395252 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23438 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1687 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,18)(8,9)(10,24)(11,16)(12,17)(13,21)(14,26)(15,28)(19,20)(22,31)(23,33)(25,27)(29,32)(30,35)(34,36);(5,7,10,21,12,22,11,29)(6,13,16,18,17,32,24,31)(8,23,25,19,15,30,26,34)(9,20,14,33,28,36,27,35);(2,4)(5,8)(6,14)(7,19)(9,24)(10,25)(11,26)(12,15)(13,20)(16,28)(17,27)(18,33)(21,30)(22,34)(23,29)(31,35)(32,36);(5,9,10,14,12,28,11,27)(6,15,16,26,17,8,24,25)(7,20,21,33,22,36,29,35)(13,30,18,34,32,23,31,19)Polycyclic presentation:
2^9|b^2=e,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302448);