Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438017 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3306 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)(9,10,12,17,13,15,20,14)(11,16,19,23,18,22,24,21);(6,7)(10,14)(12,20)(15,17)(16,21)(19,24)(22,23);(1,2,3,4)(5,6,8,7)(10,15)(14,17)(16,22)(21,23);(5,7,8,6)(9,11)(10,16)(12,19)(13,18)(14,21)(15,22)(17,23)(20,24);(11,18)(16,22)(19,24)(21,23)Polycyclic presentation:
2^9|a^2=f,c^2=ghi,d^2=h,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72342362, #72342369, #72342370, … |
| maximal normal subgroup of | #72342362, #72342369, #72342370, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372772);