Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 |
| prime radical | C2×#3055 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(9,10,13,15)(11,19,21,24)(12,18,14,17)(16,22,23,20);(1,2,4,3)(5,6)(7,8)(10,15)(16,23)(17,18)(19,24);(1,3,4,2)(6,8)(9,11,13,21)(10,16,15,23)(12,20,14,22)(17,19,18,24);(6,8);(9,12,13,14)(10,17,15,18)(11,20,21,22)(16,19,23,24)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72315432, #72315471, #72316015, … |
| maximal normal subgroup of | #72315432, #72315471, #72316015, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345967);