Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| 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 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,19,22,18,24,17,23)(15,25,26,20,16,21,27,28);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,18)(17,26)(19,27)(21,23)(22,25)(24,28);(2,4)(6,9)(10,12)(13,15,18,16)(14,21,24,25)(17,27,19,26)(20,22,28,23);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,15)(14,21,24,25)(17,26,19,27)(20,22,28,23);(13,17,18,19)(14,22,24,23)(15,26,16,27)(20,25,28,21)Polycyclic presentation:
2^9|a^2=fg,c^2=i,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6357128);