Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60454 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3155 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,8,7)(10,14)(12,19)(15,20)(16,21)(18,23)(22,24);(9,10,12,14,13,16,19,21)(11,15,18,20,17,22,23,24);(1,2,3,4)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24);(5,7,8,6)(10,16)(14,21)(15,22)(20,24);(11,17)(15,22)(18,23)(20,24)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,d^b=di,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72216137, #72224689, #72224762, … |
| maximal normal subgroup of | #72216137, #72224689, #72224762, … |
| maximal quotient of | #72216058, #72216602, #72224780, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280967);