Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3287 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,18,19)(15,23,24,20)(16,21,25,27)(17,22,26,28);(1,2,4,3)(6,9)(10,12)(14,19)(15,20)(17,22)(21,27)(23,24)(26,28);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,15)(14,20)(16,17)(18,24)(19,23)(21,28)(22,27)(25,26);(13,16,18,25)(14,21,19,27)(15,22,24,28)(17,23,26,20);(13,17,18,26)(14,22,19,28)(15,16,24,25)(20,27,23,21)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=fi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6486440);