Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #34964 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3191 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)(7,10)(8,9)(11,12)(13,14,18,24)(15,21,16,27)(17,22,19,23)(20,25,28,26);(1,2,3,4)(13,15,17,26,18,16,19,25)(14,20,23,21,24,28,22,27);(6,9)(10,12)(13,16,18,15)(14,21,24,27)(17,26,19,25)(20,22,28,23);(13,15,18,16)(14,21,24,27)(17,25,19,26)(20,22,28,23);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,22,24,23)(15,25,16,26)(20,21,28,27)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=i,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361932);