Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994043 |
| 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 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2269 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,22)(16,20)(17,26)(18,21)(19,25)(23,28)(24,27);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(15,23)(19,27)(22,28)(24,25);(5,7,8,12)(6,9,10,11)(13,15,18,23)(14,19,21,27)(16,25,17,24)(20,28,26,22);(2,4)(7,12)(9,11)(13,16,18,17)(14,20,21,26)(15,24,23,25)(19,22,27,28);(6,10)(9,11)(14,21)(15,23)(20,26)(24,25)Polycyclic presentation:
2^9|c^2=h,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6961273);