Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15258 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3306 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)(5,6,9,12)(7,11,8,10)(13,14,17,19)(15,16,20,18)(21,22,25,27)(23,24,28,26);(5,7,9,8)(6,10,12,11)(13,15,17,20)(14,18,19,16)(21,23,25,28)(22,26,27,24);(7,8)(10,11)(14,19)(16,18)(21,22,25,27)(23,26,28,24);(1,2)(3,4)(5,8,9,7)(6,11,12,10)(13,16,17,18)(14,15,19,20)(21,22)(23,26)(24,28)(25,27);(5,6)(7,11)(8,10)(9,12)(13,15)(14,16)(17,20)(18,19)(21,24,25,26)(22,23,27,28)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fgh,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,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, 2991744);