Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2487 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1044 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)(5,6,12,17,11,16,10,15)(7,18,30,13,22,32,21,33)(8,23,27,9,26,14,25,29)(19,34,24,28,31,20,35,36);(5,7,12,21,11,22,10,30)(6,13,17,18,16,33,15,32)(8,24,27,19,26,35,25,31)(9,28,14,34,29,36,23,20);(5,8)(6,14)(7,19)(9,15)(10,25)(11,26)(12,27)(13,28)(16,23)(17,29)(18,34)(20,32)(21,35)(22,31)(24,30)(33,36);(2,4)(5,9,10,23,11,29,12,14)(6,8,15,25,16,26,17,27)(7,20,21,28,22,34,30,36)(13,31,18,24,33,19,32,35)Polycyclic presentation:
2^9|a^2=eh,b^2=ef,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72322778 |
| maximal normal subgroup of | #72322778 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357319);