Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2966 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6,8,7)(9,10,13,17)(11,14,19,16)(12,20,21,22)(15,23,24,18);(1,2)(3,4)(6,7)(9,11,13,19)(10,14,17,16)(12,18,21,23)(15,22,24,20);(2,4)(5,6,8,7)(9,12,11,18,13,21,19,23)(10,15,16,20,17,24,14,22);(2,4)(5,7,8,6);(12,21)(15,24)(18,23)(20,22)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72282950, #72283418, #72283450, … |
| maximal normal subgroup of | #72282950, #72283418, #72283450, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351444);