Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#3513 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(13,14,20,24)(15,19,23,28)(16,17,27,21)(18,22,26,25);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,15,20,23)(14,19,24,28)(16,25,27,22)(17,18,21,26);(5,7)(6,9)(8,12)(10,11)(13,16,20,27)(14,21,24,17)(15,25,23,22)(18,19,26,28);(5,6)(7,10)(8,11)(9,12)(13,17,20,21)(14,16,24,27)(15,18,23,26)(19,25,28,22);(5,6)(7,9)(8,11)(10,12)(13,18)(14,22)(15,21)(16,19)(17,23)(20,26)(24,25)(27,28);(5,7)(6,10)(8,12)(9,11)(13,19)(14,23)(15,24)(16,26)(17,25)(18,27)(20,28)(21,22)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,f^e=fhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7567975);