Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
4336 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)(7,8)(9,10)(11,12)(13,14)(15,19)(16,20)(17,18)(22,23)(24,28);(6,10)(7,11)(13,15)(14,16)(17,20)(18,19)(22,23)(26,27);(6,10)(8,12)(14,18)(15,20)(21,22,25,23)(24,26,28,27);(1,2)(3,4)(5,7,9,11)(6,8,10,12)(13,16,17,19)(14,15,18,20)(21,23,25,22)(24,27,28,26);(5,8)(6,11)(7,10)(9,12)(13,15)(14,19)(16,18)(17,20)(21,24)(22,26)(23,27)(25,28)Polycyclic presentation:
2^9|c^2=fg,d^2=g,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 | #72328935, #72329042, #72329084, … |
| maximal normal subgroup of | #72328935, #72329042, #72329084, … |
| maximal quotient of | #72330207, #72333045 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4505264);