Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23444 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10)(11,17,19,14)(12,15,20,23)(13,16)(18,21)(22,24);(5,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,24)(21,23);(2,4)(6,8)(9,12)(10,15,16,23)(11,18)(13,20)(14,21,17,22)(19,24);(12,20)(15,23)(18,24)(21,22)Polycyclic presentation:
2^9|a^2=h,c^2=e,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^b=hi,d^c=di,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72265319, #72265370, #72265387, … |
| maximal normal subgroup of | #72265319, #72265370, #72265387, … |
| maximal quotient of | #72266008, #72266041 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265090);