Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422233 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,15)(8,20)(9,24)(10,12)(11,18)(13,23)(14,25)(16,27)(17,28)(19,30)(21,26)(22,29)(31,36)(32,35)(33,34);(1,2,3,4)(5,7,10,19)(6,11,12,27)(8,21,23,31)(9,16,25,18)(13,28,20,34)(14,30,24,15)(17,32,33,22)(26,35,36,29);(2,4)(6,12)(7,16,19,18)(11,15,27,30)(13,20)(14,24)(17,31,33,21)(26,28,36,34)(29,35);(5,8,9,22,10,23,25,32)(6,13,14,29,12,20,24,35)(7,17,18,21,19,33,16,31)(11,26,15,28,27,36,30,34)Polycyclic presentation:
2^9|b^2=eg,c^2=fi,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 345531);