Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,8,9)(7,10,11,12)(13,14,17,18)(15,16,20,19)(22,26)(23,29)(28,32)(31,35);(7,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,22,25,26)(23,30,29,27)(24,32,33,28)(31,34,35,36);(6,9)(10,12)(14,18)(15,20)(21,22)(23,27)(24,32)(25,26)(28,33)(29,30)(31,36)(34,35);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,18)(17,20)(21,23,25,29)(22,27,26,30)(24,31,33,35)(28,34,32,36);(5,7)(6,10)(8,11)(9,12)(13,16,17,19)(14,15,18,20)(21,24,25,33)(22,28,26,32)(23,31,29,35)(27,34,30,36)Polycyclic presentation:
2^9|a^2=fg,b^2=f,d^2=fgi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,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, 5188742);