Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455826 |
| 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)(7,15)(8,12)(9,22,25,13)(10,18)(11,14)(16,20)(17,24,32,27)(19,33,34,28)(21,30)(23,31,35,36)(26,29);(1,2)(3,4)(5,7,15,6,11,18,10,14)(8,16,26,30,21,29,20,12)(9,17,24,13,25,32,27,22)(19,31,23,33,34,36,35,28);(2,4)(5,8,11,21)(6,12,14,30)(7,16,18,29)(9,23,25,35)(10,26,15,20)(13,31,22,36)(17,33,32,28)(19,27,34,24);(5,9)(6,13)(7,17)(8,19)(10,27)(11,25)(12,28)(14,22)(15,24)(16,31)(18,32)(20,35)(21,34)(23,26)(29,36)(30,33)Polycyclic presentation:
2^9|a^2=f,b^2=efi,c^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,e^d=ei,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72285452, #72285522, #72285620, … |
| maximal normal subgroup of | #72285452, #72285522, #72285620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363839);