Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #469776 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1223 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)(7,15)(8,12)(9,22)(10,14)(11,18)(13,20)(16,27)(17,28)(19,33)(21,26)(23,32)(24,30)(25,34)(29,36)(31,35);(5,7)(6,11,14,24)(8,17)(9,18)(10,16)(12,23,26,35)(13,15,22,27)(19,32)(20,30)(21,29)(25,28,33,36)(31,34);(7,16)(8,19)(9,20)(12,25)(13,22)(15,27)(17,31)(21,34)(23,28)(26,33)(29,32)(35,36);(2,4)(5,8,20,34,10,21,9,19)(6,12,13,25,14,26,22,33)(7,17,30,31,16,29,18,32)(11,23,15,28,24,35,27,36)Polycyclic presentation:
2^9|b^2=gi,d^2=eg,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72250742, #72250767, #72250788, … |
| maximal normal subgroup of | #72250742, #72250767, #72250788, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 377479);