Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C23 | C2×C4×C8 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(13,14)(15,18,22,25)(16,19)(17,20)(21,27,28,24)(23,26);(5,6,7,9,8,10,11,12)(13,15)(14,18,20,25)(16,21)(17,22)(19,24,26,27)(23,28);(2,4)(13,16)(14,19)(15,21)(17,23)(18,24)(20,26)(22,28)(25,27)Polycyclic presentation:
2^9|a^2=f,b^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh116 conjugacy classes — view the full character table.
| maximal subgroup of | #72282597, #72282618, #72282647 |
| maximal normal subgroup of | #72282597, #72282618, #72282647 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13564);