Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×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 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5171 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)(9,10,14,18)(11,19,21,16)(12,15,22,24)(13,23,20,17);(1,2,3,4)(5,6)(7,8)(9,11,20,22,14,21,13,12)(10,15,23,16,18,24,17,19);(6,8)(9,12,13,21,14,22,20,11)(10,16,17,15,18,19,23,24);(6,8);(11,21)(12,22)(15,24)(16,19)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,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 | #72249507, #72249555, #72313499, … |
| maximal normal subgroup of | #72249507, #72249555, #72313499, … |
| maximal quotient of | #72264509, #72265295, #72268460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350443);