Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438955 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35251 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10,14,18)(11,15,16,24)(12,21,22,20)(13,19,23,17);(2,4)(5,6)(7,8)(9,11)(10,15)(12,19)(13,20)(14,16)(17,22)(18,24)(21,23);(6,8)(9,12,13,15,14,22,23,24)(10,16,17,21,18,11,19,20);(6,8);(11,16)(12,22)(15,24)(20,21)Polycyclic presentation:
2^9|a^2=i,c^2=g,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72313541, #72313664, #72313685 |
| maximal normal subgroup of | #72313541, #72313664, #72313685 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349968);