Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396118 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2572 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1011 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)(7,12,18,28)(8,20,22,13)(9,16)(10,14)(11,15)(17,33)(19,27,24,29)(21,30)(23,31,25,32)(26,34)(35,36);(2,4)(5,7,10,19,11,18,9,24)(6,12,16,29,15,28,14,27)(8,21,23,17,22,35,25,34)(13,30,31,26,20,36,32,33);(5,8)(6,13)(7,17)(9,25)(10,23)(11,22)(12,26)(14,32)(15,20)(16,31)(18,34)(19,35)(21,24)(27,30)(28,33)(29,36);(6,14,15,16)(8,22)(12,27,28,29)(13,31,20,32)(17,34)(21,35)(23,25)(26,36,33,30)Polycyclic presentation:
2^9|a^2=ef,b^2=h,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=di,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280664 |
| maximal normal subgroup of | #72280664 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 315054);