Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996607 |
| 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 | #59815 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(7,11)(10,12)(13,14,18,21)(15,19,17,26)(16,23,24,28)(20,22,27,25);(2,3)(6,9)(7,11)(13,15,18,17)(14,19,21,26)(16,22,24,25)(20,23,27,28);(5,6)(7,10)(8,9)(11,12)(13,16,15,22,18,24,17,25)(14,20,26,28,21,27,19,23);(1,3,4,2)(6,9)(10,12)(16,24)(20,27)(22,25)(23,28);(5,7,8,11)(6,10,9,12)(16,24)(20,27)(22,25)(23,28)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=gi,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7063196);