Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417540 |
| 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×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1086 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)(5,6,11,25,10,15,9,14)(7,16,24,12,19,31,18,27)(8,20,26,30,23,13,22,34)(17,32,35,36,33,29,21,28);(5,7,9,24,10,19,11,18)(6,12,14,31,15,27,25,16)(8,21,22,17,23,35,26,33)(13,29,30,28,20,32,34,36);(12,27)(16,31)(28,36)(29,32);(2,4)(5,8,9,22,10,23,11,26)(6,13,14,30,15,20,25,34)(7,17,18,21,19,33,24,35)(12,28,16,29,27,36,31,32)Polycyclic presentation:
2^9|a^2=eh,b^2=ef,d^2=e,e^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=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72252906, #72256869, #72256873 |
| maximal normal subgroup of | #72252906, #72256869, #72256873 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357316);