Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2537 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2838 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)(5,6,8,9)(7,10,12,11)(13,14)(15,18)(16,17)(19,20)(23,27)(25,28);(5,6)(7,11)(8,9)(10,12)(13,15,19,17)(14,16,20,18)(21,22,24,26)(23,25,27,28);(1,2,3,4)(6,9)(10,11)(13,16,19,18)(14,15,20,17);(5,6)(7,10)(8,9)(11,12)(13,17,19,15)(14,16,20,18)(21,23,24,27)(22,25,26,28);(5,7,8,12)(6,10,9,11)(13,18,19,16)(14,15,20,17)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=gh,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 633800);