Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6,9,11)(7,8,12,10)(13,14,16,18)(15,19,20,17)(21,22,24,25)(23,27,28,26);(5,7)(6,10)(8,11)(9,12)(13,14,16,18)(15,17,20,19)(22,25)(26,27);(6,11)(7,12)(13,14)(15,17)(16,18)(19,20)(21,23,24,28)(22,26,25,27);(1,2)(3,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,26)(24,25)(27,28);(5,7)(6,8)(9,12)(10,11)(13,15)(14,17)(16,20)(18,19)(21,23)(22,26)(24,28)(25,27)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72232524, #72232527, #72232553, … |
| maximal normal subgroup of | #72232524, #72232527, #72232553, … |
| maximal quotient of | #72298137, #72323637, #72330103, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4514757);