Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
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 | |
3640 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)(6,9)(7,11)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,27)(26,28);(5,6,8,9)(7,10,11,12)(14,18)(15,20)(21,23,25,28)(22,24,27,26);(5,7)(6,10)(8,11)(9,12)(13,15,17,20)(14,19,18,16)(21,22,25,27)(23,26,28,24);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,18)(21,23)(22,26)(24,27)(25,28);(5,7)(6,10)(8,11)(9,12)(13,16,17,19)(14,20,18,15)(21,24,25,26)(22,23,27,28)Polycyclic presentation:
2^9|b^2=fh,c^2=h,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
| maximal subgroup of | #72256722, #72256782, #72256783, … |
| maximal normal subgroup of | #72256722, #72256782, #72256783, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 757663);