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 | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3131 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)(7,9)(8,11)(10,12)(13,14,17,18)(15,16,19,20)(21,22)(23,24)(25,28)(26,27);(6,7)(8,12)(14,18)(16,20)(21,23,25,27)(22,26,28,24);(8,12)(10,11)(14,18)(16,20)(21,22,25,28)(23,24,27,26);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,15)(14,16)(17,19)(18,20)(21,24,25,26)(22,27,28,23);(5,8)(6,10)(7,11)(9,12)(13,16)(14,19)(15,18)(17,20)(21,22)(23,24)(25,28)(26,27)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,d^2=fgh,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, 3777202);