Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2617 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2853 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,7)(8,12)(13,14,18,19)(15,17,16,20)(22,26)(23,28);(5,6)(7,9)(8,10)(11,12)(13,15,18,16)(14,17,19,20)(21,22,25,26)(23,24,28,27);(6,7)(8,12)(14,19)(15,16)(23,28)(24,27);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,16,18,15)(14,17,19,20)(21,23,25,28)(22,27,26,24);(5,8)(6,10)(7,11)(9,12)(13,17)(14,15)(16,19)(18,20)(21,24,25,27)(22,28,26,23)Polycyclic presentation:
2^9|a^2=g,b^2=fh,d^2=i,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4990545);