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 | |
5333 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)(9,10,15,20)(11,21,23,17)(12,24,22,16)(13,18,14,19);(1,2,3,4)(5,6)(7,8)(9,11,13,22,15,23,14,12)(10,16,19,21,20,24,18,17);(6,8)(9,12,14,23,15,22,13,11)(10,17,18,24,20,21,19,16);(6,8)(9,13,15,14)(10,18,20,19)(11,22,23,12)(16,17,24,21);(11,23)(12,22)(16,24)(17,21)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=i,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, #72249565, … |
| maximal normal subgroup of | #72249507, #72249555, #72249565, … |
| maximal quotient of | #72264400, #72264404, #72264691, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350445);