Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
2346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,18)(15,16,19,20)(21,22)(23,27)(24,28)(25,26);(1,2,3,4)(6,9)(10,11)(13,14)(15,16)(17,18)(19,20)(22,25)(23,24);(1,2,3,4)(6,9)(10,11)(14,18)(16,20)(21,23,26,24)(22,27,25,28);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(13,15)(14,16)(17,19)(18,20)(21,24,26,23)(22,28,25,27);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,16)(14,19)(15,18)(17,20)(21,25,26,22)(23,27,24,28)Polycyclic presentation:
2^9|a^2=fgh,b^2=i,c^2=fgi,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3936430);