Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420453 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23399 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1914 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)(5,6,11,16)(7,17,20,29)(8,13,22,28)(9,15,10,14)(12,23,26,18)(19,31,34,35)(21,30,24,27)(25,33,36,32);(7,18,20,23)(12,17,26,29)(19,33,34,32)(25,35,36,31);(5,7,9,18,11,20,10,23)(6,12,14,17,16,26,15,29)(8,19,21,32,22,34,24,33)(13,25,27,31,28,36,30,35);(2,4)(5,8)(6,13)(7,19)(9,21)(10,24)(11,22)(12,25)(14,27)(15,30)(16,28)(17,31)(18,32)(20,34)(23,33)(26,36)(29,35)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72286018, #72286176, #72286206 |
| maximal normal subgroup of | #72286018, #72286176, #72286206 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330145);