Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #400051 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2535 |
| 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 | |
1421 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)(6,12)(7,17)(8,22)(9,11)(13,18)(14,28)(15,16)(19,33)(23,24)(26,35)(27,30)(31,34);(5,6)(7,18)(8,14)(9,15)(10,16)(11,12)(13,20)(17,30)(19,32)(21,27)(22,25)(23,29)(24,28)(26,33)(31,36)(34,35);(5,7)(6,13)(8,19)(9,20)(10,17)(11,21)(12,18)(14,26)(15,30)(16,27)(22,34)(23,33)(24,31)(25,32)(28,35)(29,36);(2,4)(5,8,11,22,10,24,9,23)(6,14,12,25,16,28,15,29)(7,19,21,34,17,31,20,33)(13,26,18,32,27,35,30,36)Polycyclic presentation:
2^9|d^2=ef,e^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,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 307247);