Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #428059 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| 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 | |
1177 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)(7,17)(8,22)(9,12)(10,16)(11,15)(13,20)(14,23)(18,26)(19,33)(21,28)(24,29)(25,30)(27,32)(31,35)(34,36);(1,2)(3,4)(6,12)(7,18,21,20)(8,23)(13,17,26,28)(15,16)(19,32,35,34)(24,25)(27,31,36,33);(2,4)(5,7)(6,13)(8,19)(9,21)(10,18)(11,20)(12,26)(14,27)(15,28)(16,17)(22,36)(23,35)(24,32)(25,34)(29,33)(30,31);(5,8,11,25,9,23,10,24)(6,14,16,30,12,22,15,29)(7,19,20,34,21,35,18,32)(13,27,17,31,26,36,28,33)Polycyclic presentation:
2^9|b^2=ei,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335256);