Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1361 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)(6,12,16,15)(7,17,21,20)(8,22,25,24)(14,29)(19,33)(23,30)(26,36,34,32)(31,35);(5,6)(7,18)(8,23)(9,15)(10,12)(11,16)(13,17)(14,24)(19,34)(20,28)(21,27)(22,29)(25,30)(26,33)(31,32)(35,36);(2,4)(5,7)(6,13)(8,19)(9,17)(10,20)(11,21)(12,18)(14,26)(15,27)(16,28)(22,35)(23,32)(24,31)(25,33)(29,34)(30,36);(5,8,9,24,11,25,10,22)(6,14,12,23,16,29,15,30)(7,19,17,31,21,33,20,35)(13,26,18,32,28,34,27,36)Polycyclic presentation:
2^9|a^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327268);