Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806527 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1862 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)(7,11)(8,10)(9,12);(5,7,8,12)(6,9,10,11)(13,14)(15,24)(16,20)(17,19)(18,23)(21,26)(22,25)(27,28);(1,2,3,4)(6,10)(7,12)(13,15,25,27,18,26,16,17)(14,19,20,21,23,28,22,24);(6,10)(9,11)(13,16,18,25)(14,20,23,22)(15,17,26,27)(19,21,28,24);(13,17,16,26,18,27,25,15)(14,21,22,19,23,24,20,28)Polycyclic presentation:
2^9|b^2=f,c^2=ghi,d^2=i,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6723373);