Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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×#2456 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2507 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)(7,11)(8,9)(10,12)(13,14)(15,18)(16,17)(19,20)(21,22,26,24)(23,29,30,27)(25,32,33,28)(31,34,36,35);(5,7)(6,10)(8,11)(9,12)(13,14,17,16)(15,19,20,18)(22,24)(23,30)(28,32)(31,36);(13,15)(14,18)(16,19)(17,20)(21,23,26,30)(22,27,24,29)(25,31,33,36)(28,34,32,35);(1,2)(3,4)(5,8,9,6)(7,11,12,10)(13,16,17,14)(15,19,20,18)(21,24,26,22)(23,29,30,27)(25,28,33,32)(31,34,36,35);(5,7)(6,11)(8,10)(9,12)(13,15)(14,19)(16,18)(17,20)(21,25,26,33)(22,28,24,32)(23,31,30,36)(27,34,29,35)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4769652);