Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,12)(7,11,8,10)(13,14,18,20)(15,16,17,19)(21,22,25,23)(24,27,28,26);(2,4)(5,7,9,8)(6,10,12,11)(13,15,18,17)(14,19,20,16)(21,23,25,22)(24,27,28,26);(1,2)(3,4)(5,6,9,12)(7,10,8,11)(13,16,18,19)(14,15,20,17);(5,6)(7,11)(8,10)(9,12)(13,16)(14,15)(17,20)(18,19)(21,24,25,28)(22,26,23,27);(5,8,9,7)(6,11,12,10)(13,17,18,15)(14,19,20,16)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=f,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,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, 2686565);