Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(7,11)(9,12)(13,14,17,15)(16,19,20,18)(21,22,26,27)(23,24,30,28)(25,33,35,29)(31,34,36,32);(2,4)(5,6,8,10)(7,9,11,12)(22,27)(23,30)(29,33)(31,36);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,15,17,14)(16,19,20,18)(21,23,26,30)(22,28,27,24)(25,31,35,36)(29,34,33,32);(5,6)(7,9)(8,10)(11,12)(13,16,17,20)(14,18,15,19)(21,24,26,28)(22,23,27,30)(25,34,35,32)(29,36,33,31);(5,7)(6,9)(8,11)(10,12)(21,25,26,35)(22,29,27,33)(23,31,30,36)(24,32,28,34)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=i,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2654385);