Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3010 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)(5,6,9,10)(7,11,12,8)(14,17)(15,19)(21,22)(23,27)(24,28)(25,26);(5,7,9,12)(6,8,10,11)(15,19)(18,20)(22,26)(24,27);(6,10)(7,12)(13,14)(15,18)(16,17)(19,20)(21,23,25,28)(22,27,26,24);(1,2,3,4)(5,8)(6,7)(9,11)(10,12)(13,14)(15,18)(16,17)(19,20)(21,24)(22,23)(25,27)(26,28);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,19)(17,20)(21,23)(22,27)(24,26)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fh,d^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72300368, #72300383, #72300452 |
| maximal normal subgroup of | #72300368, #72300383, #72300452 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4008332);