Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514060 |
| 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 | #2416 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2006 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)(7,10)(8,9)(11,12)(13,14,19,24)(15,20,23,18)(16,26,28,22)(17,21,27,25);(1,2)(3,4)(6,9)(10,12)(13,15,19,23)(14,18,24,20)(16,21,28,25)(17,22,27,26);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,22)(17,18)(19,28)(20,27)(23,26)(24,25);(2,4)(13,17,19,27)(14,22,24,26)(15,25,23,21)(16,18,28,20);(6,9)(10,12)(13,18,19,20)(14,23,24,15)(16,27,28,17)(21,22,25,26)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fi,d^2=i,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,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,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, 7422375);