Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #612424 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4488 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)(6,7)(8,12)(14,17)(18,20)(21,22)(23,27)(24,25)(26,28);(5,6,9,7)(8,10,12,11)(13,14)(15,18)(16,17)(19,20)(22,25)(26,27);(1,2)(3,4)(21,22)(23,26)(24,25)(27,28);(5,7,9,6)(8,11,12,10)(13,15,16,19)(14,18,17,20);(5,8)(6,10)(7,11)(9,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,26)(24,28)(25,27)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch116 conjugacy classes — view the full character table.
| maximal subgroup of | #72267287, #72329669, #72329681, … |
| maximal normal subgroup of | #72267287, #72329669, #72329681, … |
| maximal quotient of | #72267316, #72331558, #72336202 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 521947);