Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2078 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10,13,16)(11,18,21,24)(12,15,22,17)(14,19,23,20);(2,4)(9,11)(10,14)(12,19)(13,21)(15,18)(16,23)(17,24)(20,22);(10,15)(11,19)(12,22)(16,17)(18,24)(20,21);(5,7,8,6)(10,16)(14,23)(15,17)(18,24)Polycyclic presentation:
2^9|a^2=h,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,e^c=ei,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224689, #72226922, #72226928, … |
| maximal normal subgroup of | #72224689, #72226922, #72226928, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266217);