Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455828 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,16,11,15,10,13)(7,12,20,27,19,25,17,28)(8,21,22,30,24,29,23,14)(18,32,33,26,34,36,31,35);(1,2)(3,4)(5,7,10,20,11,19,9,17)(6,12,16,28,15,25,13,27)(8,18,23,33,24,34,22,31)(14,26,21,32,30,35,29,36);(2,4)(6,13)(7,17)(9,10)(12,25)(14,21)(15,16)(18,33)(19,20)(22,23)(26,35)(29,30)(31,34);(5,8)(6,14)(7,18)(9,22)(10,23)(11,24)(12,26)(13,29)(15,30)(16,21)(17,31)(19,34)(20,33)(25,35)(27,36)(28,32)Polycyclic presentation:
2^9|a^2=f,b^2=efi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=de,f^c=fi,g^c=gi,h^c=hi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72285452, #72285534, #72285592 |
| maximal normal subgroup of | #72285452, #72285534, #72285592 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363862);