Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418535 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3041 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1279 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)(5,6,10,14)(7,17)(8,23)(9,15,27,16)(11,22)(12,25)(13,26)(18,31,34,36)(19,28,35,29)(20,33)(21,30)(24,32);(5,7,9,21,10,22,27,20)(6,11,15,33,14,17,16,30)(8,19,25,34,26,35,24,18)(12,31,13,28,32,36,23,29);(2,4)(5,8)(6,12,14,32)(7,18)(9,24)(10,26)(11,28,17,29)(13,15,23,16)(19,20)(21,35)(22,34)(25,27)(30,36,33,31);(5,8)(6,13,14,23)(7,19)(9,25)(10,26)(11,29,17,28)(12,15,32,16)(18,20)(21,34)(22,35)(24,27)(30,36,33,31)Polycyclic presentation:
2^9|a^2=efg,b^2=eg,c^2=e,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,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, 348420);