Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #382440 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2309 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14338 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,10,19,26)(3,11,14,20,13,25,18,7)(4,8,16,23,17,24,31,32)(12,27,28,22,15,29,30,21);(7,20)(11,25)(21,22)(27,29);(1,3)(2,7)(4,15,17,12)(5,18)(6,13)(8,22,24,21)(9,25)(10,20)(11,26)(14,19)(16,28,31,30)(23,27,32,29);(1,4)(2,8)(3,12)(5,16)(6,17)(7,21)(9,23)(10,24)(11,27)(13,15)(14,28)(18,30)(19,31)(20,22)(25,29)(26,32)Polycyclic presentation:
2^9|a^2=f,c^2=h,f^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^c=de,e^c=ei,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72240457, #72240458, #72240461, … |
| maximal normal subgroup of | #72240457, #72240458, #72240461, … |
| maximal quotient of | #72256382 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289746);