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 | |
2940 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(13,14)(15,21)(16,24)(17,22)(18,19)(20,26)(23,27)(25,28);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,18)(14,19,22,21)(16,25,26,23)(20,28,24,27);(2,4)(14,19)(15,18)(20,27)(21,22)(23,25)(24,28);(6,10)(9,11)(13,16)(14,20)(15,23)(17,26)(18,25)(19,27)(21,28)(22,24);(2,4)(7,12)(9,11)(13,15,17,18)(14,21,22,19)(16,25,26,23)(20,27,24,28)Polycyclic presentation:
2^9|b^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, 6575281);