Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,10)(7,11,12,9)(13,14)(15,24)(16,20)(17,21)(18,22)(19,23)(25,28)(26,27);(5,7,8,12)(6,9,10,11)(15,25)(16,26)(17,19)(20,27)(21,23)(24,28);(1,2,3,4)(13,15)(14,20)(16,18)(17,25)(19,26)(21,27)(22,24)(23,28);(13,16)(14,20)(15,18)(17,26)(19,25)(21,27)(22,24)(23,28);(13,17,18,19)(14,21,22,23)(15,25,16,26)(20,27,24,28)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350591);