Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453981 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(9,10)(11,12)(13,14,19,24)(15,25,16,28)(17,23,18,22)(20,27,21,26);(5,6)(7,8)(13,15,19,16)(14,20,24,21)(17,27,18,26)(22,28,23,25);(2,4)(6,8)(15,26)(16,27)(17,18)(20,25)(21,28)(22,23);(10,12)(13,16,19,15)(14,21,24,20)(17,26,18,27)(22,25,23,28)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72315713, #72315717, #72315737, … |
| maximal normal subgroup of | #72315713, #72315717, #72315737, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361185);