Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #25109 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,12)(8,10)(9,11)(13,14,19,22)(15,17,21,18)(16,23,26,28)(20,27,24,25);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,17)(16,24)(18,22)(19,21)(20,26)(23,25)(27,28);(5,8,9,7)(6,10,11,12)(13,16,19,26)(14,20,22,24)(15,23,21,28)(17,25,18,27);(2,4)(13,16)(14,20)(15,23)(17,25)(18,27)(19,26)(21,28)(22,24);(6,11)(10,12)(13,17,19,18)(14,21,22,15)(16,25,26,27)(20,28,24,23)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,g^c=gi,h^c=hi,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7422949);