Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#392 |
| 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 | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3446 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)(7,11)(8,9)(10,12)(13,14,17,18)(15,16,19,20)(21,22,25,27)(23,24,28,26);(2,4)(13,15)(14,16)(17,19)(18,20)(21,23,25,28)(22,26,27,24);(1,2,3,4)(6,9)(10,11)(14,18)(15,19)(21,23,25,28)(22,26,27,24);(5,7)(6,10)(8,12)(9,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,26)(27,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,28)(26,27)Polycyclic presentation:
2^9|a^2=fh,b^2=fgi,c^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^c=eg110 conjugacy classes — view the full character table.
| maximal subgroup of | #72303671, #72303678, #72303690, … |
| maximal normal subgroup of | #72303671, #72303678, #72303690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4213984);