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 | |
3486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,11)(8,9)(10,12)(13,14)(15,16)(17,18)(19,20)(21,22,24,26)(23,25,28,27);(1,2)(3,4)(14,18)(16,19)(21,22,24,26)(23,25,28,27);(2,3)(6,9)(10,11)(21,22)(23,27)(24,26)(25,28);(5,7)(6,10)(8,12)(9,11)(13,15)(14,19)(16,18)(17,20);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,16,17,19)(14,15,18,20)(21,23)(22,25)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=h,b^2=fgi,e^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, #72303676, #72303687, … |
| maximal normal subgroup of | #72303671, #72303676, #72303687, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4213995);