Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #988785 |
| automorphism | ? |
| 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 | C2×#2668 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,9,10)(7,8,11,12)(13,14)(15,18)(16,17)(19,20)(21,22,24,25)(23,27,28,26);(1,2,4,3)(6,10)(7,11)(13,14,17,16)(15,19,20,18)(22,25)(26,27);(2,3)(6,10)(8,12)(14,16)(18,19)(21,23,24,28)(22,26,25,27);(5,7)(6,8)(9,11)(10,12)(13,15)(14,18)(16,19)(17,20)(21,22)(23,26)(24,25)(27,28);(1,3,4,2)(5,8)(6,7)(9,12)(10,11)(13,16,17,14)(15,19,20,18)(21,23)(22,26)(24,28)(25,27)Polycyclic presentation:
2^9|a^2=fh,b^2=h,c^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72234646, #72234678, #72260758, … |
| maximal normal subgroup of | #72234646, #72234678, #72260758, … |
| maximal quotient of | #72328949, #72329202 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 950902);