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 | |
2069 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)(13,14,18,26,17,21,15,19)(16,23,22,20,25,27,24,28);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15,17,18)(14,19,21,26)(16,22,25,24)(20,27,28,23);(5,7,8,12)(6,9,10,11)(13,16)(14,20)(15,22)(17,25)(18,24)(19,27)(21,28)(23,26);(2,4)(7,12)(9,11)(14,21)(19,26)(20,28)(23,27);(6,10)(9,11)(13,15,17,18)(14,19,21,26)(16,22,25,24)(20,27,28,23)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6969540);