Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2758 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)(7,11)(8,10)(9,12)(13,14)(15,18)(16,23)(17,21)(19,22)(20,26)(24,27)(25,28);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,18)(16,24)(17,22)(19,21)(20,26)(23,27)(25,28);(2,4)(13,16,19,27)(14,20,22,28)(15,23,21,24)(17,26,18,25);(6,10)(9,11)(13,14)(15,17)(16,20)(18,21)(19,22)(23,26)(24,25)(27,28);(2,4)(7,12)(9,11)(13,17,19,18)(14,21,22,15)(16,25,27,26)(20,23,28,24)Polycyclic presentation:
2^9|c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6575277);